{VERSION 5 0 "Windows 10" "5.0" } {USTYLETAB {PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 4 2 0 2 0 2 2 -1 1 }{PSTYLE "War ning" -1 7 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Left Justified Maple Outp ut" -1 12 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Fixed Width" -1 17 1 {CSTYLE " " -1 -1 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Help" -1 10 1 {CSTYLE "" -1 -1 "Courier" 1 9 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Head ing 4" -1 20 1 {CSTYLE "" -1 -1 "Times" 1 10 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Line Printed Output" -1 6 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "T imes" 1 14 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }1 1 0 0 8 2 2 0 2 0 2 2 -1 1 } {PSTYLE "Maple Output" -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Heading 3" -1 5 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Diagnostic" -1 9 1 {CSTYLE "" -1 -1 "Courier" 1 10 64 128 64 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 1" -1 200 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Text Output" -1 2 1 {CSTYLE "" -1 -1 "Courier" 1 10 0 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Ordered List 2" -1 201 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 36 2 0 2 2 -1 1 }{PSTYLE "Ordered List 3" -1 202 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 72 2 0 2 2 -1 1 }{PSTYLE "Ordered List 4" -1 203 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 108 2 0 2 2 -1 1 } {PSTYLE "Ordered List 5" -1 204 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 144 2 0 2 2 -1 1 }{PSTYLE "Annota tion Title" -1 205 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 2 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "HyperlinkError" -1 206 1 {CSTYLE "" -1 -1 "Courier New" 1 12 255 0 255 1 2 2 1 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "HyperlinkWarning" -1 207 1 {CSTYLE "" -1 -1 "Courier New" 1 12 0 0 255 1 2 2 1 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Bullet Item" -1 15 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Maple Plot" -1 13 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "List \+ Item" -1 14 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Dash Item" -1 16 1 {CSTYLE " " -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 3 3 2 0 2 0 2 2 -1 1 }{PSTYLE "Error" -1 8 1 {CSTYLE "" -1 -1 "Courier" 1 10 255 0 255 1 2 2 2 2 2 1 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Tit le" -1 18 1 {CSTYLE "" -1 -1 "Times" 1 18 0 0 0 1 2 1 1 2 2 2 1 0 0 1 }3 1 0 0 12 12 2 0 2 0 2 2 -1 1 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }1 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "Author" -1 19 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 8 8 2 0 2 0 2 2 -1 1 }{CSTYLE "Help Variab le" -1 25 "Courier" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple C omment" -1 21 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Pop up" -1 31 "Times" 1 12 0 128 128 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Atomi c Variable" -1 200 "Times" 1 12 175 0 175 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "Plot Text" -1 28 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 } {CSTYLE "Code" -1 201 "Courier New" 1 12 255 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic" -1 3 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Small" -1 7 "Times" 1 1 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold Small" -1 10 "Times" 1 1 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Bold" -1 39 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Menus" -1 36 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Normal" -1 30 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input" -1 0 "Courier" 1 12 255 0 0 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "Header and Footer" -1 202 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Output Labels" -1 29 "Times" 1 8 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small202" -1 203 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Notes" -1 37 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined" -1 44 "Times" 1 12 0 0 0 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Text" -1 204 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Plot Title" -1 27 "Times" 1 10 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Default" -1 38 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Italic Bold" -1 40 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Maple Input Placeholder" -1 205 "C ourier New" 1 12 200 0 200 1 2 1 2 2 1 2 0 0 0 1 }{CSTYLE "LaTeX" -1 32 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Emphasized" -1 22 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Equation Lab el" -1 206 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Ita lic" -1 42 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "Prompt" -1 1 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Bold " -1 5 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Hyperlink" -1 17 "Times" 1 12 0 128 128 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "Caption T ext" -1 207 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Un derlined Bold" -1 41 "Times" 1 12 0 0 0 1 1 1 2 2 2 2 0 0 0 1 }{CSTYLE "Copyright" -1 34 "Times" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE " Dictionary Hyperlink" -1 45 "Times" 1 12 147 0 15 1 2 2 1 2 2 2 0 0 0 1 }{CSTYLE "2D Input" -1 19 "Times" 1 12 255 0 0 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Help Heading" -1 26 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Output" -1 20 "Times" 1 12 0 0 255 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Inert Output" -1 208 "Times" 1 12 144 144 144 1 2 2 2 2 1 2 0 0 0 1 }{CSTYLE "Page Number" -1 33 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Fixed" -1 23 "Courier" 1 10 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Italic Small" -1 209 "Times" 1 1 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math Symbol 2" -1 16 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Nonterminal" -1 24 "Courier" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Annotation Text" -1 210 "Time s" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{CSTYLE "Help Maple Name" -1 35 " Times" 1 12 104 64 92 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "Help Underlined \+ Italic" -1 43 "Times" 1 12 0 0 0 1 1 2 1 2 2 2 0 0 0 1 }{CSTYLE "Capti on Reference" -1 211 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "2D Comment" -1 18 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 208 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 209 1 {CSTYLE "" -1 -1 "T imes" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {CSTYLE "" -1 212 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 213 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 214 "T imes" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 215 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 216 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 217 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 218 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 219 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 220 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 221 "T imes" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 222 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 223 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 224 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 225 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 226 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 227 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 228 "T imes" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 210 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{PSTYLE "" -1 211 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 229 "Tim es" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 230 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 231 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 212 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 }{CSTYLE "" -1 232 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 233 "Time s" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 234 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 235 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 236 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 237 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 } {CSTYLE "" -1 238 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 239 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 240 "T imes" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 241 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{PSTYLE "" -1 213 1 {CSTYLE "" -1 -1 "Ti mes" 1 12 0 0 0 1 2 2 2 2 2 2 1 0 0 1 }3 1 0 0 0 0 2 0 2 0 2 2 -1 1 } {CSTYLE "" -1 242 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 243 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 244 "T imes" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 245 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 0 0 0 1 }{CSTYLE "" -1 246 "Times" 1 12 0 0 0 1 1 2 2 2 2 2 0 0 0 1 }} {SECT 0 {EXCHG {PARA 208 "" 0 "" {TEXT 204 0 "" }}}{EXCHG {PARA 209 " " 0 "" {TEXT 204 27 "Setting up Double Integrals" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 247 267 "We will begin by drawing a region and showing how the geometry of the region leads to the set-up of an iterated integra l. The operative phrase is \223curve-to-curve, point-to-point\224. As we promised in an earlier session, we will begin with a region that i s bounded by " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\" ), Typesetting:-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvar iant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", f ence = \"false\", separator = \"false\", stretchy = \"false\", symmetr ic = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typ esetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), Typ esetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sys libGF'6%-I#miGF$6#Q!F'-F#6%-F,6%Q\"yF'/%'italicGQ%trueF'/%,mathvariant GQ'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separator GFB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'acc entGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F,6%Q\"xF'F4F7F+" }{TEXT 247 4 " , " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), \+ Typesetting:-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvarian t = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fenc e = \"false\", separator = \"false\", stretchy = \"false\", symmetric \+ = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typese tting:-mrow(Typesetting:-mn(\"2\", mathvariant = \"normal\"), Typesett ing:-mo(\"−\", mathvariant = \"normal\", fence = \"false\", sepa rator = \"false\", stretchy = \"false\", symmetric = \"false\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mi(\"x\", it alic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\")), T ypesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_s yslibGF'6%-I#miGF$6#Q!F'-F#6&-F,6%Q\"yF'/%'italicGQ%trueF'/%,mathvaria ntGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separat orGFB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'a ccentGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F#6%-I#mnGF$6$Q\"2F'F>-F ;6-Q(−F'F>F@FCFEFGFIFKFM/FPQ,0.2222222emF'/FSFhn-F,6%Q\"xF'F4F7F +F+" }{TEXT 247 4 " , " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting :-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"true\" , mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"no rmal\", fence = \"false\", separator = \"false\", stretchy = \"false\" , symmetric = \"false\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778 em\"), Typesetting:-mn(\"0\", mathvariant = \"normal\")), Typesetting: -mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%- I#miGF$6#Q!F'-F#6%-F,6%Q\"xF'/%'italicGQ%trueF'/%,mathvariantGQ'italic F'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/%)str etchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB/%' lspaceGQ,0.2777778emF'/%'rspaceGFQ-I#mnGF$6$Q\"0F'F>F+" }{TEXT 247 156 " . The first two lines of Maple syntax produce a plot showing th e region. The next four lines produce a plot to help explain why the \+ region requires that " }{TEXT 212 10 "dA = dy dx" }{TEXT 247 2 " ." }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart: with(plots):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "plot([x,2-x],x=0..1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "region:=plot([x,2-x],x=0..1, color=red): with(plottools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "column:=curve([[.38,.4],[.42,.4],[.42,1.6],[.38,1.6],[.38,.4]],c olor=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "element:=cu rve([[.38,.7],[.38,.74],[.42,.74],[.42,.7],[.38,.7]],color=blue):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "display(region,column,elemen t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 247 30 "If you top this region wi th " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typese tting:-mrow(Typesetting:-mi(\"z\", italic = \"true\", mathvariant = \" italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \" false\", separator = \"false\", stretchy = \"false\", symmetric = \"fa lse\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting: -mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-msup(Types etting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typese tting:-mn(\"2\", mathvariant = \"normal\"), superscriptshift = \"0\")) , Typesetting:-mo(\"−\", mathvariant = \"normal\", fence = \"fal se\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mr ow(Typesetting:-msup(Typesetting:-mi(\"y\", italic = \"true\", mathvar iant = \"italic\"), Typesetting:-mn(\"2\", mathvariant = \"normal\"), \+ superscriptshift = \"0\")), Typesetting:-mi(\"\")), Typesetting:-mi(\" \")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesetti ngGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6&-F,6%Q\"zF'/%'italicGQ%trueF'/%,ma thvariantGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%* separatorGFB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimits GFB/%'accentGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F#6'F+-F#6#-I%msu pGF$6%-F,6%Q\"xF'F4F7-I#mnGF$6$Q\"2F'F>/%1superscriptshiftGQ\"0F'-F;6- Q(−F'F>F@FCFEFGFIFKFM/FPQ,0.2222222emF'/FSFco-F#6#-FY6%-F,6%Q\"y F'F4F7FhnF\\oF+F+F+" }{TEXT 247 59 " , then you get the figure produc ed by the commands below." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "top:=plot3d([x,y,4-x^2-y^2],y=x..2-x,x=0..1):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 45 "side1:=plot3d([x,x,z],z=0..4-x^2-x^2,x=0..1) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "side2:=plot3d([x,2-x,z ],z=0..4-x^2-(2-x)^2,x=0..1):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "side3:=plot3d([0,y,z],z=0..4-0^2-y^2,y=0..2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "display(top,side1,side2,side3,axes=BOXED); " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 247 90 "If we draw horizontal lines \+ across the region we see that the lines will always start at " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\" ), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", lsp ace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mn(\"0 \", mathvariant = \"normal\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I +modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%-F,6%Q\" xF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'nor malF'/%&fenceGQ&falseF'/%*separatorGFB/%)stretchyGFB/%*symmetricGFB/%( largeopGFB/%.movablelimitsGFB/%'accentGFB/%'lspaceGQ,0.2777778emF'/%'r spaceGFQ-I#mnGF$6$Q\"0F'F>F+" }{TEXT 247 16 " , which is the " }{TEXT 213 1 "y" }{TEXT 247 50 "-axis, but they can end on two different curv es, " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typese tting:-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \" italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \" false\", separator = \"false\", stretchy = \"false\", symmetric = \"fa lse\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting: -mi(\"x\", italic = \"true\", mathvariant = \"italic\")), Typesetting: -mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%- I#miGF$6#Q!F'-F#6%-F,6%Q\"yF'/%'italicGQ%trueF'/%,mathvariantGQ'italic F'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/%)str etchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB/%' lspaceGQ,0.2777778emF'/%'rspaceGFQ-F,6%Q\"xF'F4F7F+" }{TEXT 247 7 " a nd " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typeset ting:-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"i talic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"f alse\", separator = \"false\", stretchy = \"false\", symmetric = \"fal se\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:- mrow(Typesetting:-mn(\"2\", mathvariant = \"normal\"), Typesetting:-mo (\"−\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"f alse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.22 22222em\", rspace = \"0.2222222em\"), Typesetting:-mi(\"x\", italic = \+ \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\")), Typesett ing:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF '6%-I#miGF$6#Q!F'-F#6&-F,6%Q\"yF'/%'italicGQ%trueF'/%,mathvariantGQ'it alicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/% )stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGF B/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F#6%-I#mnGF$6$Q\"2F'F>-F;6-Q(&m inus;F'F>F@FCFEFGFIFKFM/FPQ,0.2222222emF'/FSFhn-F,6%Q\"xF'F4F7F+F+" } {TEXT 247 60 " . This would mean that integrating first with respect to " }{TEXT 214 1 "x" }{TEXT 247 172 " , which changes from left to r ight, would violate the \221curve-to-curve\222 concept for our inside \+ integral. As you see in the right-hand plot, every vertical line begi ns at " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Type setting:-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \+ \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesettin g:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), Typesettin g:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6 %-I#miGF$6#Q!F'-F#6%-F,6%Q\"yF'/%'italicGQ%trueF'/%,mathvariantGQ'ital icF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/%)s tretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB/ %'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F,6%Q\"xF'F4F7F+" }{TEXT 247 15 " and ends on " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\" \"), Typesetting:-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathv ariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symme tric = \"false\", largeop = \"false\", movablelimits = \"false\", acce nt = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), T ypesetting:-mrow(Typesetting:-mn(\"2\", mathvariant = \"normal\"), Typ esetting:-mo(\"−\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", lsp ace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mi(\"x \", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\" \")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesetti ngGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6&-F,6%Q\"yF'/%'italicGQ%trueF'/%,ma thvariantGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%* separatorGFB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimits GFB/%'accentGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F#6%-I#mnGF$6$Q\" 2F'F>-F;6-Q(−F'F>F@FCFEFGFIFKFM/FPQ,0.2222222emF'/FSFhn-F,6%Q\"x F'F4F7F+F+" }{TEXT 247 15 " . Think of " }{TEXT 215 2 "y " }{TEXT 247 92 ", which increases by moving up, and you see that the inside in tegral can be with respect to " }{TEXT 216 1 "y" }{TEXT 247 277 " . V isualize the small rectangle as it slides up the longer one. Now, ima gine that little rectangle as the base of a vertical column inside the solid. As the volume of each column is added up along the long recta ngle we obtain the volume of a \223slab\224 that is parallel to the " }{TEXT 217 2 "yz" }{TEXT 247 454 "-plane. Think of the first integrat ion as determining the volumes of all those slabs. The second integra tion adds up the volumes of the slabs to get the volume of the solid. \+ At what number does the \221first\222 slab occur? At what number does the \221last\222 slab occur? That is the \223point-to-point\224 part of the process. This idea is illustrated in the plots that follow. \+ First, a value is established for each column along the strip that is \+ parallel to the " }{TEXT 218 1 "y" }{TEXT 247 213 "-axis. The diagram produced by the commands below outlines the column in red together wi th the edges of the region. It is important here to realize that the \+ values for each column are summed between two curves, " }{TEXT 219 6 " y=g(x)" }{TEXT 247 7 " and " }{TEXT 220 6 "y=h(x)" }{TEXT 247 36 ", \+ as the first step in the process.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "c1:=spacecurve([0,y,4-y^2],y=0..2,axes=NORMAL,color=b lack):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "c2:=spacecurve(\{ [x,2-x,4-x^2-(2-x)^2],[x,x,4-2*x^2]\},x=0..1,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "c3:=spacecurve(\{[x,x,0],[x,2-x,0], [1,1,2*x]\},x=0..1,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "c4:=spacecurve(\{[.38,y,4-.38^2-y^2],[.42,y,4-.42^2-y^2]\},y= .4..1.6,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 143 "c 5:=curve([[.38,.4,3.68],[.42,.4,3.68],[.42,.4,0],[.42,1.6,0],[.42,1.6, 1.28],[.38,1.6,1.28],[.38,1.6,0],[.38,.4,0],[.38,.4,3.68]],color=black ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 232 "col:=curve([[.38,.7, 0],[.38,.7,3.32],[.42,.7,3.32],[.42,.7,0],[.38,.7,0],[.38,.74,0],[.38, .74,3.32],[.42,.74,3.32],[.42,.74,0],[.38,.74,0],[.42,.74,0],[.42,.7,0 ],[.42,.7,3.32],[.42,.74,3.32],[.38,.74,3.32],[.38,.7,3.32]],color=red ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "display(c1,c2,c3,c4,c 5,col);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 247 166 "The next step is to \+ add up the values for each \223slab\224 and obtain a value for the sol id. This addition occurs between two constants, or points. Here the \+ points are " }{TEXT 221 5 "x = 0" }{TEXT 247 8 " and " }{TEXT 222 6 "x = 1 " }{TEXT 247 50 ". Visualize this as beginning with a slab i n the " }{TEXT 223 2 "yz" }{TEXT 247 13 "-plane where " }{TEXT 224 1 " x" }{TEXT 247 1 " " }{TEXT 225 4 "= 0 " }{TEXT 247 35 "and stacking sl abs in the positive " }{TEXT 226 1 "x" }{TEXT 247 45 "-direction until you reach the value (point) " }{TEXT 227 5 "x =1 " }{TEXT 247 106 ". \+ Executing the following Maple commands will color in the surfaces of \+ the slab from the previous figure." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "s1:=plot3d([.42,y,z],z=0..4-.42^2-y^2,y=.4..1.6):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "s2:=plot3d([.38,y,z],z=0..4 -.38^2-y^2,y=.4..1.6):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "s 3:=plot3d([x,1.6,z],z=0..1.28,x=.38..0.42):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "s4:=plot3d([x,.4,z],z=0..3.68,x=.38..0.42):" }} {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "s5:=plot3d([x,y,0],y=.4..1.6,x=.38. .0.42):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "s6:=plot3d([x,y, 4-.4^2-y^2],y=.4..1.6,x=.38..0.42):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "display(c1,c2,c3,s1,s2,s3,s4,s5,s6,col);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 247 0 "" }}{PARA 0 "" 0 "" {TEXT 247 90 "We are \+ ready to start the set-up process. We know that we must integrate wit h respect to " }{TEXT 228 1 "y" }{TEXT 247 133 " first. We must comp lete this iterated integral in the format shown, filling in the \"blan ks\" in the limits of integration as shown." }}}{EXCHG {PARA 210 "" 0 "" {TEXT 204 4 " " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-m i(\"\"), Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mo(\"&int ;\", foreground = \"[144,144,144]\", mathvariant = \"normal\", Typeset ting:-msemantics = \"inert\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspac e = \"0.0em\"), Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"tr ue\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \+ \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777 778em\"), Typesetting:-mi(\"blank\", italic = \"true\", mathvariant = \+ \"italic\")), Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"true \", mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \" normal\", fence = \"false\", separator = \"false\", stretchy = \"false \", symmetric = \"false\", largeop = \"false\", movablelimits = \"fals e\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.277777 8em\"), Typesetting:-mi(\"blank\", italic = \"true\", mathvariant = \" italic\")), superscriptshift = \"2\", subscriptshift = \"0\"), Typeset ting:-mi(\"\"), Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mo (\"∫\", foreground = \"[144,144,144]\", mathvariant = \"normal\", \+ Typesetting:-msemantics = \"inert\", fence = \"false\", separator = \" false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\" , rspace = \"0.0em\"), Typesetting:-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvar iant = \"normal\", fence = \"false\", separator = \"false\", stretchy \+ = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \+ \"0.2777778em\"), Typesetting:-mi(\"blank\", italic = \"true\", mathva riant = \"italic\")), Typesetting:-mrow(Typesetting:-mi(\"y\", italic \+ = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvari ant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits \+ = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \" 0.2777778em\"), Typesetting:-mi(\"blank\", italic = \"true\", mathvari ant = \"italic\")), superscriptshift = \"2\", subscriptshift = \"0\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mn(\"4\", mathvar iant = \"normal\"), Typesetting:-mo(\"−\", mathvariant = \"norma l\", fence = \"false\", separator = \"false\", stretchy = \"false\", s ymmetric = \"false\", largeop = \"false\", movablelimits = \"false\", \+ accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\" ), Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mn(\"2\", mathvaria nt = \"normal\"), superscriptshift = \"0\")), Typesetting:-mo(\"&minus ;\", mathvariant = \"normal\", fence = \"false\", separator = \"false \", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", \+ movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em \", rspace = \"0.2222222em\"), Typesetting:-mrow(Typesetting:-msup(Typ esetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Type setting:-mn(\"2\", mathvariant = \"normal\"), superscriptshift = \"0\" )), Typesetting:-mi(\"\")), mathvariant = \"normal\"), Typesetting:-mi (\"\"), Typesetting:-mspace(height = \"0.0ex\", width = \"0.3em\", dep th = \"0.0ex\", linebreak = \"auto\"), Typesetting:-mo(\"&Differential D;\", foreground = \"[144,144,144]\", mathvariant = \"normal\", Typese tting:-msemantics = \"inert\", fence = \"false\", separator = \"false \", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", \+ movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rsp ace = \"0.0em\"), Typesetting:-mi(\"y\", italic = \"true\", mathvarian t = \"italic\")), Typesetting:-mi(\"\"), Typesetting:-mspace(height = \+ \"0.0ex\", width = \"0.3em\", depth = \"0.0ex\", linebreak = \"auto\") , Typesetting:-mo(\"ⅆ\", foreground = \"[144,144,144]\", mathvariant = \"normal\", Typesetting:-msemantics = \"inert\", fence \+ = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+ \"false\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x \", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\" \"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF $6#Q!F'-F#6)-I(msubsupGF$6'-I#moGF$6/Q&∫F'/%+foregroundGQ.[144,144 ,144]F'/%,mathvariantGQ'normalF'/I+msemanticsGF$Q&inertF'/%&fenceGQ&fa lseF'/%*separatorGFC/%)stretchyGFC/%*symmetricGFC/%(largeopGFC/%.movab lelimitsGFC/%'accentGFC/%'lspaceGQ&0.0emF'/%'rspaceGFR-F#6%-F,6%Q\"xF' /%'italicGQ%trueF'/FFAFDFFFHFJFL FNFPFSF^pF+F\\rFjrFWF+" }{TEXT 204 5 " = " }{XPPEDIT 18 0 "Typesetti ng:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-msubsup (Typesetting:-mo(\"∫\", foreground = \"[144,144,144]\", mathvarian t = \"normal\", Typesetting:-msemantics = \"inert\", fence = \"false\" , separator = \"false\", stretchy = \"false\", symmetric = \"false\", \+ largeop = \"false\", movablelimits = \"false\", accent = \"false\", ls pace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow(Typesetting:- mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-m o(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"fa lse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.277777 8em\", rspace = \"0.2777778em\"), Typesetting:-mi(\"a\", italic = \"tr ue\", mathvariant = \"italic\")), Typesetting:-mrow(Typesetting:-mi(\" x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"= \", mathvariant = \"normal\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mi(\"b\", italic = \"true\", \+ mathvariant = \"italic\")), superscriptshift = \"2\", subscriptshift = \"0\"), Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-msubsup (Typesetting:-mo(\"∫\", foreground = \"[144,144,144]\", mathvarian t = \"normal\", Typesetting:-msemantics = \"inert\", fence = \"false\" , separator = \"false\", stretchy = \"false\", symmetric = \"false\", \+ largeop = \"false\", movablelimits = \"false\", accent = \"false\", ls pace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow(Typesetting:- mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-m o(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"fa lse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.277777 8em\", rspace = \"0.2777778em\"), Typesetting:-mrow(Typesetting:-mi(\" g\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"& ApplyFunction;\", mathvariant = \"normal\", fence = \"false\", separat or = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \" 0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mrow(T ypesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), m athvariant = \"normal\")), Typesetting:-mi(\"\")), Typesetting:-mrow(T ypesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Ty pesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", sepa rator = \"false\", stretchy = \"false\", symmetric = \"false\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mrow(Typeset ting:-mi(\"h\", italic = \"true\", mathvariant = \"italic\"), Typesett ing:-mo(\"⁡\", mathvariant = \"normal\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesett ing:-mrow(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"it alic\")), mathvariant = \"normal\")), Typesetting:-mi(\"\")), superscr iptshift = \"2\", subscriptshift = \"0\"), Typesetting:-mfenced(Typese tting:-mrow(Typesetting:-mn(\"4\", mathvariant = \"normal\"), Typesett ing:-mo(\"−\", mathvariant = \"normal\", fence = \"false\", sepa rator = \"false\", stretchy = \"false\", symmetric = \"false\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typeset ting:-msup(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"i talic\"), Typesetting:-mn(\"2\", mathvariant = \"normal\"), superscrip tshift = \"0\")), Typesetting:-mo(\"−\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\") , Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"y\", italic = \+ \"true\", mathvariant = \"italic\"), Typesetting:-mn(\"2\", mathvarian t = \"normal\"), superscriptshift = \"0\")), Typesetting:-mi(\"\")), m athvariant = \"normal\"), Typesetting:-mi(\"\"), Typesetting:-mspace(h eight = \"0.0ex\", width = \"0.3em\", depth = \"0.0ex\", linebreak = \+ \"auto\"), Typesetting:-mo(\"ⅆ\", foreground = \"[144,14 4,144]\", mathvariant = \"normal\", Typesetting:-msemantics = \"inert \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesettin g:-mi(\"y\", italic = \"true\", mathvariant = \"italic\")), Typesettin g:-mi(\"\"), Typesetting:-mspace(height = \"0.0ex\", width = \"0.3em\" , depth = \"0.0ex\", linebreak = \"auto\"), Typesetting:-mo(\"&Differe ntialD;\", foreground = \"[144,144,144]\", mathvariant = \"normal\", T ypesetting:-msemantics = \"inert\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x\", italic = \"true\", mathva riant = \"italic\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenam eG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6)-I(msubsupGF$6'-I# moGF$6/Q&∫F'/%+foregroundGQ.[144,144,144]F'/%,mathvariantGQ'normal F'/I+msemanticsGF$Q&inertF'/%&fenceGQ&falseF'/%*separatorGFC/%)stretch yGFC/%*symmetricGFC/%(largeopGFC/%.movablelimitsGFC/%'accentGFC/%'lspa ceGQ&0.0emF'/%'rspaceGFR-F#6%-F,6%Q\"xF'/%'italicGQ%trueF'/FFAFDFFFHFJFLFNFPFSFcpF+FdsFbtFWF+" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 247 0 "" }}{PARA 0 "" 0 "" {TEXT 247 42 " This means that the inner integral will be" }}}{EXCHG {PARA 211 "" 0 " " {TEXT 204 2 " " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi( \"\"), Typesetting:-mrow(Typesetting:-msubsup(Typesetting:-mo(\"∫ \", foreground = \"[144,144,144]\", mathvariant = \"normal\", Typesett ing:-msemantics = \"inert\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mov ablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow(Typesetting:-mi(\"y\", italic = \"tru e\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \+ \"normal\", fence = \"false\", separator = \"false\", stretchy = \"fal se\", symmetric = \"false\", largeop = \"false\", movablelimits = \"fa lse\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777 778em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"it alic\")), Typesetting:-mrow(Typesetting:-mi(\"y\", italic = \"true\", \+ mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"norm al\", fence = \"false\", separator = \"false\", stretchy = \"false\", \+ symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em \"), Typesetting:-mrow(Typesetting:-mn(\"2\", mathvariant = \"normal\" ), Typesetting:-mo(\"−\", mathvariant = \"normal\", fence = \"fa lse\", separator = \"false\", stretchy = \"false\", symmetric = \"fals e\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:- mi(\"x\", italic = \"true\", mathvariant = \"italic\")), Typesetting:- mi(\"\")), superscriptshift = \"2\", subscriptshift = \"0\"), Typesett ing:-mfenced(Typesetting:-mrow(Typesetting:-mn(\"4\", mathvariant = \" normal\"), Typesetting:-mo(\"−\", mathvariant = \"normal\", fenc e = \"false\", separator = \"false\", stretchy = \"false\", symmetric \+ = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typese tting:-mrow(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"true\" , mathvariant = \"italic\"), Typesetting:-mn(\"2\", mathvariant = \"no rmal\"), superscriptshift = \"0\")), Typesetting:-mo(\"−\", math variant = \"normal\", fence = \"false\", separator = \"false\", stretc hy = \"false\", symmetric = \"false\", largeop = \"false\", movablelim its = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace \+ = \"0.2222222em\"), Typesetting:-mrow(Typesetting:-msup(Typesetting:-m i(\"y\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mn (\"2\", mathvariant = \"normal\"), superscriptshift = \"0\")), Typeset ting:-mi(\"\")), mathvariant = \"normal\"), Typesetting:-mi(\"\"), Typ esetting:-mspace(height = \"0.0ex\", width = \"0.3em\", depth = \"0.0e x\", linebreak = \"auto\"), Typesetting:-mo(\"ⅆ\", foreg round = \"[144,144,144]\", mathvariant = \"normal\", Typesetting:-msem antics = \"inert\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimit s = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0e m\"), Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic \")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesetti ngGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6(-I(msubsupGF$6'-I#moGF$6/Q&∫F' /%+foregroundGQ.[144,144,144]F'/%,mathvariantGQ'normalF'/I+msemanticsG F$Q&inertF'/%&fenceGQ&falseF'/%*separatorGFC/%)stretchyGFC/%*symmetric GFC/%(largeopGFC/%.movablelimitsGFC/%'accentGFC/%'lspaceGQ&0.0emF'/%'r spaceGFR-F#6%-F,6%Q\"yF'/%'italicGQ%trueF'/FFAFDFFFHFJFLFNFP FSFWF+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 247 80 "Think of a vertical li ne sliding from left to right over the region. The first " }{TEXT 229 3 " x " }{TEXT 247 23 " value that occurs is " }{TEXT 230 5 "x = \+ 0" }{TEXT 247 18 " and the last is " }{TEXT 231 4 "x =1" }{TEXT 247 34 ". So our double integral becomes\n" }{MPLTEXT 1 0 1 " " }}} {EXCHG {PARA 212 "" 0 "" {TEXT 204 2 " " }{XPPEDIT 18 0 "Typesetting: -mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-msubsup(Ty pesetting:-mo(\"∫\", foreground = \"[144,144,144]\", mathvariant = \"normal\", Typesetting:-msemantics = \"inert\", fence = \"false\", s eparator = \"false\", stretchy = \"false\", symmetric = \"false\", lar geop = \"false\", movablelimits = \"false\", accent = \"false\", lspac e = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow(Typesetting:-mi( \"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo( \"=\", mathvariant = \"normal\", fence = \"false\", separator = \"fals e\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em \", rspace = \"0.2777778em\"), Typesetting:-mn(\"0\", mathvariant = \" normal\")), Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"true\" , mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"no rmal\", fence = \"false\", separator = \"false\", stretchy = \"false\" , symmetric = \"false\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778 em\"), Typesetting:-mn(\"1\", mathvariant = \"normal\")), superscripts hift = \"2\", subscriptshift = \"0\"), Typesetting:-mi(\"\"), Typesett ing:-mrow(Typesetting:-msubsup(Typesetting:-mo(\"∫\", foreground = \"[144,144,144]\", mathvariant = \"normal\", Typesetting:-msemantics \+ = \"inert\", fence = \"false\", separator = \"false\", stretchy = \"fa lse\", symmetric = \"false\", largeop = \"false\", movablelimits = \"f alse\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), T ypesetting:-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typese tting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), Typese tting:-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \" italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \" false\", separator = \"false\", stretchy = \"false\", symmetric = \"fa lse\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting: -mrow(Typesetting:-mn(\"2\", mathvariant = \"normal\"), Typesetting:-m o(\"−\", mathvariant = \"normal\", fence = \"false\", separator \+ = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \" false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2 222222em\", rspace = \"0.2222222em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\")), supersc riptshift = \"2\", subscriptshift = \"0\"), Typesetting:-mfenced(Types etting:-mrow(Typesetting:-mn(\"4\", mathvariant = \"normal\"), Typeset ting:-mo(\"−\", mathvariant = \"normal\", fence = \"false\", sep arator = \"false\", stretchy = \"false\", symmetric = \"false\", large op = \"false\", movablelimits = \"false\", accent = \"false\", lspace \+ = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mrow(Typese tting:-msup(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \" italic\"), Typesetting:-mn(\"2\", mathvariant = \"normal\"), superscri ptshift = \"0\")), Typesetting:-mo(\"−\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\") , Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi(\"y\", italic = \+ \"true\", mathvariant = \"italic\"), Typesetting:-mn(\"2\", mathvarian t = \"normal\"), superscriptshift = \"0\")), Typesetting:-mi(\"\")), m athvariant = \"normal\"), Typesetting:-mi(\"\"), Typesetting:-mspace(h eight = \"0.0ex\", width = \"0.3em\", depth = \"0.0ex\", linebreak = \+ \"auto\"), Typesetting:-mo(\"ⅆ\", foreground = \"[144,14 4,144]\", mathvariant = \"normal\", Typesetting:-msemantics = \"inert \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesettin g:-mi(\"y\", italic = \"true\", mathvariant = \"italic\")), Typesettin g:-mi(\"\"), Typesetting:-mspace(height = \"0.0ex\", width = \"0.3em\" , depth = \"0.0ex\", linebreak = \"auto\"), Typesetting:-mo(\"&Differe ntialD;\", foreground = \"[144,144,144]\", mathvariant = \"normal\", T ypesetting:-msemantics = \"inert\", fence = \"false\", separator = \"f alse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x\", italic = \"true\", mathva riant = \"italic\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenam eG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6)-I(msubsupGF$6'-I# moGF$6/Q&∫F'/%+foregroundGQ.[144,144,144]F'/%,mathvariantGQ'normal F'/I+msemanticsGF$Q&inertF'/%&fenceGQ&falseF'/%*separatorGFC/%)stretch yGFC/%*symmetricGFC/%(largeopGFC/%.movablelimitsGFC/%'accentGFC/%'lspa ceGQ&0.0emF'/%'rspaceGFR-F#6%-F,6%Q\"xF'/%'italicGQ%trueF'/FFAFDF FFHFJFLFNFPFSFcpF+FdrFbsFWF+" }{TEXT 204 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 204 98 "BEWARE! In Maple,we must be sure to put the inner in tegral limits first and the outer ones last.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "with(student): A:=Doubleint(4-x^2-y^2,y=x..(2-x ),x=0..1);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(A);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "int(int(4-x^2-y^2,y=x..2-x ),x=0..1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 247 232 "Incidentally, the value of this integral is 8/3 . Compare the syntax for the double in tegral to the equivalent non-parametric syntax we would have used to p lot the top in the graphing section (below). The similarities are obv ious." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "plot3d(4-x^2-y^2,y =x..(2-x),x=0..1,color=blue,axes=boxed);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 247 0 "" }}{PARA 0 "" 0 "" {TEXT 232 9 "Example: " }{TEXT 247 30 " Sketch the solid bounded by " }{XPPEDIT 18 0 "Typesetting:-mrow( Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvar iant = \"normal\", fence = \"false\", separator = \"false\", stretchy \+ = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \+ \"0.2777778em\"), Typesetting:-mrow(Typesetting:-msup(Typesetting:-mi( \"y\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mn( \"2\", mathvariant = \"normal\"), superscriptshift = \"0\")), Typesett ing:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I ,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6&-F,6%Q\"xF'/%'italicGQ%t rueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ& falseF'/%*separatorGFB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.mov ablelimitsGFB/%'accentGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F#6#-I% msupGF$6%-F,6%Q\"yF'F4F7-I#mnGF$6$Q\"2F'F>/%1superscriptshiftGQ\"0F'F+ F+" }{TEXT 247 3 " , " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:- mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", \+ mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"norm al\", fence = \"false\", separator = \"false\", stretchy = \"false\", \+ symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em \"), Typesetting:-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathv ariant = \"italic\"), Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symme tric = \"false\", largeop = \"false\", movablelimits = \"false\", acce nt = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), T ypesetting:-mn(\"2\", mathvariant = \"normal\")), Typesetting:-mi(\"\" )), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesetting GI(_syslibGF'6%-I#miGF$6#Q!F'-F#6&-F,6%Q\"xF'/%'italicGQ%trueF'/%,math variantGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*se paratorGFB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGF B/%'accentGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F#6%-F,6%Q\"yF'F4F7 -F;6-Q\"+F'F>F@FCFEFGFIFKFM/FPQ,0.2222222emF'/FSFgn-I#mnGF$6$Q\"2F'F>F +F+" }{TEXT 247 3 " , " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting: -mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"z\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"nor mal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\" , accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em \"), Typesetting:-mn(\"0\", mathvariant = \"normal\")), Typesetting:-m i(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I# miGF$6#Q!F'-F#6%-F,6%Q\"zF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF' -I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/%)stret chyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB/%'ls paceGQ,0.2777778emF'/%'rspaceGFQ-I#mnGF$6$Q\"0F'F>F+" }{TEXT 247 3 " , " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesettin g:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"x\" , italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", s tretchy = \"false\", symmetric = \"false\", largeop = \"false\", movab lelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rs pace = \"0.2222222em\"), Typesetting:-mi(\"z\", italic = \"true\", mat hvariant = \"italic\")), Typesetting:-mo(\"=\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\") , Typesetting:-mn(\"4\", mathvariant = \"normal\")), Typesetting:-mi( \"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#mi GF$6#Q!F'-F#6&F+-F#6%-F,6%Q\"xF'/%'italicGQ%trueF'/%,mathvariantGQ'ita licF'-I#moGF$6-Q\"+F'/F:Q'normalF'/%&fenceGQ&falseF'/%*separatorGFD/%) stretchyGFD/%*symmetricGFD/%(largeopGFD/%.movablelimitsGFD/%'accentGFD /%'lspaceGQ,0.2222222emF'/%'rspaceGFS-F,6%Q\"zF'F6F9-F=6-Q\"=F'F@FBFEF GFIFKFMFO/FRQ,0.2777778emF'/FUFgn-I#mnGF$6$Q\"4F'F@F+" }{TEXT 247 22 " and find its volume." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 247 13 "Elimin ating " }{TEXT 233 1 "x" }{TEXT 247 39 " from the first two equation s yields " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), T ypesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting: -msup(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic \"), Typesetting:-mn(\"2\", mathvariant = \"normal\"), superscriptshif t = \"0\")), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesettin g:-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"ital ic\"), Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \"fals e\", separator = \"false\", stretchy = \"false\", symmetric = \"false \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mn (\"2\", mathvariant = \"normal\")), Typesetting:-mi(\"\")), Typesettin g:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6 %-I#miGF$6#Q!F'-F#6'F+-F#6#-I%msupGF$6%-F,6%Q\"yF'/%'italicGQ%trueF'/% ,mathvariantGQ'italicF'-I#mnGF$6$Q\"2F'/F=Q'normalF'/%1superscriptshif tGQ\"0F'-I#moGF$6-Q\"=F'FC/%&fenceGQ&falseF'/%*separatorGFN/%)stretchy GFN/%*symmetricGFN/%(largeopGFN/%.movablelimitsGFN/%'accentGFN/%'lspac eGQ,0.2777778emF'/%'rspaceGFgn-F#6%F6-FI6-Q\"+F'FCFLFOFQFSFUFWFY/FfnQ, 0.2222222emF'/FinF`oF?F+F+" }{TEXT 247 3 " , " }{XPPEDIT 18 0 "Typeset ting:-mrow(Typesetting:-mn(\"0\", mathvariant = \"normal\"));" "-I%mro wG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6#-I#mnGF$6$Q\"0F'/%,ma thvariantGQ'normalF'" }{TEXT 247 3 " = " }{XPPEDIT 18 0 "Typesetting:- mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), T ypesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting: -msup(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic \"), Typesetting:-mn(\"2\", mathvariant = \"normal\"), superscriptshif t = \"0\")), Typesetting:-mo(\"−\", mathvariant = \"normal\", fe nce = \"false\", separator = \"false\", stretchy = \"false\", symmetri c = \"false\", largeop = \"false\", movablelimits = \"false\", accent \+ = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Type setting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Types etting:-mo(\"−\", mathvariant = \"normal\", fence = \"false\", s eparator = \"false\", stretchy = \"false\", symmetric = \"false\", lar geop = \"false\", movablelimits = \"false\", accent = \"false\", lspac e = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mn(\"2\", mathvariant = \"normal\")), Typesetting:-mo(\"=\", mathvariant = \"no rmal\", fence = \"false\", separator = \"false\", stretchy = \"false\" , symmetric = \"false\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778 em\"), Typesetting:-mrow(Typesetting:-mfenced(Typesetting:-mrow(Typese tting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Typeset ting:-mo(\"−\", mathvariant = \"normal\", fence = \"false\", sep arator = \"false\", stretchy = \"false\", symmetric = \"false\", large op = \"false\", movablelimits = \"false\", accent = \"false\", lspace \+ = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mn(\"2\", m athvariant = \"normal\")), mathvariant = \"normal\"), Typesetting:-mo( \"⁡\", mathvariant = \"normal\", fence = \"false\", sepa rator = \"false\", stretchy = \"false\", symmetric = \"false\", largeo p = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mfenced(Typesetting:-mro w(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"y\", ital ic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"+\", mathv ariant = \"normal\", fence = \"false\", separator = \"false\", stretch y = \"false\", symmetric = \"false\", largeop = \"false\", movablelimi ts = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mn(\"1\", mathvariant = \"normal\")), \+ Typesetting:-mi(\"\")), mathvariant = \"normal\")), Typesetting:-mi(\" \")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesetti ngGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6'F+-F#6(F+-F#6#-I%msupGF$6%-F,6%Q\" yF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#mnGF$6$Q\"2F'/F?Q'nor malF'/%1superscriptshiftGQ\"0F'-I#moGF$6-Q(−F'FE/%&fenceGQ&false F'/%*separatorGFP/%)stretchyGFP/%*symmetricGFP/%(largeopGFP/%.movablel imitsGFP/%'accentGFP/%'lspaceGQ,0.2222222emF'/%'rspaceGFinF8FJFA-FK6-Q \"=F'FEFNFQFSFUFWFYFen/FhnQ,0.2777778emF'/F[oF`o-F#6%-I(mfencedGF$6$-F #6%F8FJFAFE-FK6-Q0⁡F'FEFNFQFSFUFWFYFen/FhnQ&0.0emF'/F[oF ]p-Feo6$-F#6%F+-F#6%F8-FK6-Q\"+F'FEFNFQFSFUFWFYFenFgnFjn-FB6$Q\"1F'FEF +FEF+F+" }{TEXT 247 68 " , so the vertical parabolic surface meets t he vertical plane at " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting: -mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"nor mal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\" , accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em \"), Typesetting:-mrow(Typesetting:-mo(\"&uminus0;\", mathvariant = \" normal\", fence = \"false\", separator = \"false\", stretchy = \"false \", symmetric = \"false\", largeop = \"false\", movablelimits = \"fals e\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0.222222 2em\"), Typesetting:-mn(\"1\", mathvariant = \"normal\")), Typesetting :-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Ty pesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6&-F,6%Q\"yF'/%'italicGQ%true F'/%,mathvariantGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&fal seF'/%*separatorGFB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movabl elimitsGFB/%'accentGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F#6$-F;6-Q *&uminus0;F'F>F@FCFEFGFIFKFM/FPQ,0.2222222emF'/FSFZ-I#mnGF$6$Q\"1F'F>F +F+" }{TEXT 247 7 " and " }{TEXT 234 1 "y" }{TEXT 247 30 " = 2 . Th e horizontal plane " }{TEXT 235 1 "z" }{TEXT 247 52 " = 0 will serve as a bottom and the oblique plane " }{XPPEDIT 18 0 "Typesetting:-mro w(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Type setting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \+ \"italic\"), Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesettin g:-mi(\"z\", italic = \"true\", mathvariant = \"italic\")), Typesettin g:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \+ \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fa lse\", movablelimits = \"false\", accent = \"false\", lspace = \"0.277 7778em\", rspace = \"0.2777778em\"), Typesetting:-mn(\"4\", mathvarian t = \"normal\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6 \"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6&F+-F#6%-F,6%Q\"xF'/%' italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q\"+F'/F:Q'normalF'/ %&fenceGQ&falseF'/%*separatorGFD/%)stretchyGFD/%*symmetricGFD/%(largeo pGFD/%.movablelimitsGFD/%'accentGFD/%'lspaceGQ,0.2222222emF'/%'rspaceG FS-F,6%Q\"zF'F6F9-F=6-Q\"=F'F@FBFEFGFIFKFMFO/FRQ,0.2777778emF'/FUFgn-I #mnGF$6$Q\"4F'F@F+" }{TEXT 247 138 " as a top. The domain in the pla ne is easy to plot parametrically. The output of the commands below s hows the base of the solid in the " }{TEXT 236 2 "xy" }{TEXT 247 7 "-p lane." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "A1:=plot([y^2,y,y= -1..2],color=red):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "A2:=p lot([y+2,y,y=-1..2],color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(A1,A2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 247 87 "We could set up our iterated integral now, but let\222s plot the s olid first. We can use " }{TEXT 237 2 "A1" }{TEXT 247 5 " and " } {TEXT 238 2 "A2" }{TEXT 247 68 " to help us. The commands below draw \+ all but the base of the solid." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "A3:=plot3d([y^2,y,z],y=-1..2,z=0..4-y^2,color=red):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "A4:=plot3d([y+2,y,z],y=-1..2 ,z=0..(2-y),color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "A5:=plot3d([x,y,4-x],x=y^2..(y+2),y=-1..2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "display(A3,A4,A5,axes=BOXED);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 247 235 "Consider horizontal and vertical lines \+ and how they would meet the region. Vertical lines could go from one \+ side of the parabola to the other or from the line to the parabola, su ggesting that one should not integrate with respect to " }{TEXT 239 1 "y" }{TEXT 247 109 " first. Horizontal lines would go from the par abola to the line. Ahah! We can integrate with respect to " }{TEXT 240 1 "x" }{TEXT 247 46 " first. The integrand is found by solving \+ " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesettin g:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"x\" , italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", s tretchy = \"false\", symmetric = \"false\", largeop = \"false\", movab lelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rs pace = \"0.2222222em\"), Typesetting:-mi(\"z\", italic = \"true\", mat hvariant = \"italic\")), Typesetting:-mo(\"=\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\") , Typesetting:-mn(\"4\", mathvariant = \"normal\")), Typesetting:-mi( \"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#mi GF$6#Q!F'-F#6&F+-F#6%-F,6%Q\"xF'/%'italicGQ%trueF'/%,mathvariantGQ'ita licF'-I#moGF$6-Q\"+F'/F:Q'normalF'/%&fenceGQ&falseF'/%*separatorGFD/%) stretchyGFD/%*symmetricGFD/%(largeopGFD/%.movablelimitsGFD/%'accentGFD /%'lspaceGQ,0.2222222emF'/%'rspaceGFS-F,6%Q\"zF'F6F9-F=6-Q\"=F'F@FBFEF GFIFKFMFO/FRQ,0.2777778emF'/FUFgn-I#mnGF$6$Q\"4F'F@F+" }{TEXT 247 7 " \+ for " }{TEXT 241 1 "z" }{TEXT 247 24 " , and that means that " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"z\", italic = \"true\", mathvariant = \"italic\" ), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", lsp ace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mrow(Ty pesetting:-mn(\"4\", mathvariant = \"normal\"), Typesetting:-mo(\"&min us;\", mathvariant = \"normal\", fence = \"false\", separator = \"fals e\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em \", rspace = \"0.2222222em\"), Typesetting:-mi(\"x\", italic = \"true \", mathvariant = \"italic\")), Typesetting:-mi(\"\")), Typesetting:-m i(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I# miGF$6#Q!F'-F#6&-F,6%Q\"zF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF' -I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/%)stret chyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB/%'ls paceGQ,0.2777778emF'/%'rspaceGFQ-F#6%-I#mnGF$6$Q\"4F'F>-F;6-Q(−F 'F>F@FCFEFGFIFKFM/FPQ,0.2222222emF'/FSFhn-F,6%Q\"xF'F4F7F+F+" }{TEXT 247 58 " . We show the three stages of setting up the integral.\n" } }}{EXCHG {PARA 213 "" 0 "" {TEXT 204 1 " " }{XPPEDIT 18 0 "Typesetting :-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-msubsup(T ypesetting:-mo(\"∫\", foreground = \"[144,144,144]\", mathvariant \+ = \"normal\", Typesetting:-msemantics = \"inert\", fence = \"false\", \+ separator = \"false\", stretchy = \"false\", symmetric = \"false\", la rgeop = \"false\", movablelimits = \"false\", accent = \"false\", lspa ce = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow(Typesetting:-mi (\"y\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo( \"=\", mathvariant = \"normal\", fence = \"false\", separator = \"fals e\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em \", rspace = \"0.2777778em\"), Typesetting:-mi(\"blank\", italic = \"t rue\", mathvariant = \"italic\")), Typesetting:-mrow(Typesetting:-mi( \"y\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo( \"=\", mathvariant = \"normal\", fence = \"false\", separator = \"fals e\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em \", rspace = \"0.2777778em\"), Typesetting:-mi(\"blank\", italic = \"t rue\", mathvariant = \"italic\")), superscriptshift = \"2\", subscript shift = \"0\"), Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:- msubsup(Typesetting:-mo(\"∫\", foreground = \"[144,144,144]\", mat hvariant = \"normal\", Typesetting:-msemantics = \"inert\", fence = \" false\", separator = \"false\", stretchy = \"false\", symmetric = \"fa lse\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow(Typese tting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typeset ting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0 .2777778em\", rspace = \"0.2777778em\"), Typesetting:-mi(\"blank\", it alic = \"true\", mathvariant = \"italic\")), Typesetting:-mrow(Typeset ting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesett ing:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator \+ = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \" false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2 777778em\", rspace = \"0.2777778em\"), Typesetting:-mi(\"blank\", ital ic = \"true\", mathvariant = \"italic\")), superscriptshift = \"2\", s ubscriptshift = \"0\"), Typesetting:-mfenced(Typesetting:-mrow(Typeset ting:-mn(\"4\", mathvariant = \"normal\"), Typesetting:-mo(\"−\" , mathvariant = \"normal\", fence = \"false\", separator = \"false\", \+ stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mova blelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", r space = \"0.2222222em\"), Typesetting:-mi(\"x\", italic = \"true\", ma thvariant = \"italic\")), mathvariant = \"normal\"), Typesetting:-mi( \"\"), Typesetting:-mspace(height = \"0.0ex\", width = \"0.3em\", dept h = \"0.0ex\", linebreak = \"auto\"), Typesetting:-mo(\"&DifferentialD ;\", foreground = \"[144,144,144]\", mathvariant = \"normal\", Typeset ting:-msemantics = \"inert\", fence = \"false\", separator = \"false\" , stretchy = \"false\", symmetric = \"false\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspac e = \"0.0em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant \+ = \"italic\")), Typesetting:-mi(\"\"), Typesetting:-mspace(height = \" 0.0ex\", width = \"0.3em\", depth = \"0.0ex\", linebreak = \"auto\"), \+ Typesetting:-mo(\"ⅆ\", foreground = \"[144,144,144]\", m athvariant = \"normal\", Typesetting:-msemantics = \"inert\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\")) ;" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q !F'-F#6)-I(msubsupGF$6'-I#moGF$6/Q&∫F'/%+foregroundGQ.[144,144,144 ]F'/%,mathvariantGQ'normalF'/I+msemanticsGF$Q&inertF'/%&fenceGQ&falseF '/%*separatorGFC/%)stretchyGFC/%*symmetricGFC/%(largeopGFC/%.movableli mitsGFC/%'accentGFC/%'lspaceGQ&0.0emF'/%'rspaceGFR-F#6%-F,6%Q\"yF'/%'i talicGQ%trueF'/FFAFDF FFHFJFLFNFPFSF^pF+F`qF^rFWF+" }{TEXT 204 7 " " }{XPPEDIT 18 0 "T ypesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting: -msubsup(Typesetting:-mo(\"∫\", foreground = \"[144,144,144]\", ma thvariant = \"normal\", Typesetting:-msemantics = \"inert\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow(Type setting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Types etting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separat or = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \" 0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mi(\"blank\", i talic = \"true\", mathvariant = \"italic\")), Typesetting:-mrow(Typese tting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Typeset ting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0 .2777778em\", rspace = \"0.2777778em\"), Typesetting:-mi(\"blank\", it alic = \"true\", mathvariant = \"italic\")), superscriptshift = \"2\", subscriptshift = \"0\"), Typesetting:-mi(\"\"), Typesetting:-mrow(Typ esetting:-msubsup(Typesetting:-mo(\"∫\", foreground = \"[144,144,1 44]\", mathvariant = \"normal\", Typesetting:-msemantics = \"inert\", \+ fence = \"false\", separator = \"false\", stretchy = \"false\", symmet ric = \"false\", largeop = \"false\", movablelimits = \"false\", accen t = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-m row(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\" ), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", lsp ace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mrow(Ty pesetting:-msup(Typesetting:-mi(\"y\", italic = \"true\", mathvariant \+ = \"italic\"), Typesetting:-mn(\"2\", mathvariant = \"normal\"), super scriptshift = \"0\")), Typesetting:-mi(\"\")), Typesetting:-mrow(Types etting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typese tting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separato r = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0 .2777778em\", rspace = \"0.2777778em\"), Typesetting:-mrow(Typesetting :-mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Typesetting: -mo(\"+\", mathvariant = \"normal\", fence = \"false\", separator = \" false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", lspace = \"0.22222 22em\", rspace = \"0.2222222em\"), Typesetting:-mn(\"2\", mathvariant \+ = \"normal\")), Typesetting:-mi(\"\")), superscriptshift = \"2\", subs criptshift = \"0\"), Typesetting:-mfenced(Typesetting:-mrow(Typesettin g:-mn(\"4\", mathvariant = \"normal\"), Typesetting:-mo(\"−\", m athvariant = \"normal\", fence = \"false\", separator = \"false\", str etchy = \"false\", symmetric = \"false\", largeop = \"false\", movable limits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspa ce = \"0.2222222em\"), Typesetting:-mi(\"x\", italic = \"true\", mathv ariant = \"italic\")), mathvariant = \"normal\"), Typesetting:-mi(\"\" ), Typesetting:-mspace(height = \"0.0ex\", width = \"0.3em\", depth = \+ \"0.0ex\", linebreak = \"auto\"), Typesetting:-mo(\"ⅆ\", foreground = \"[144,144,144]\", mathvariant = \"normal\", Typesetting :-msemantics = \"inert\", fence = \"false\", separator = \"false\", st retchy = \"false\", symmetric = \"false\", largeop = \"false\", movabl elimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \+ \"0.0em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \" italic\")), Typesetting:-mi(\"\"), Typesetting:-mspace(height = \"0.0e x\", width = \"0.3em\", depth = \"0.0ex\", linebreak = \"auto\"), Type setting:-mo(\"ⅆ\", foreground = \"[144,144,144]\", mathv ariant = \"normal\", Typesetting:-msemantics = \"inert\", fence = \"fa lse\", separator = \"false\", stretchy = \"false\", symmetric = \"fals e\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"y\", it alic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F' -F#6)-I(msubsupGF$6'-I#moGF$6/Q&∫F'/%+foregroundGQ.[144,144,144]F' /%,mathvariantGQ'normalF'/I+msemanticsGF$Q&inertF'/%&fenceGQ&falseF'/% *separatorGFC/%)stretchyGFC/%*symmetricGFC/%(largeopGFC/%.movablelimit sGFC/%'accentGFC/%'lspaceGQ&0.0emF'/%'rspaceGFR-F#6%-F,6%Q\"yF'/%'ital icGQ%trueF'/FFAFDFFFHFJFLFNFPFSF^pF+F_rF]sFWF+" }{TEXT 204 7 " " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-msubsup(Typesetting:-mo(\"∫\", foreground = \"[14 4,144,144]\", mathvariant = \"normal\", Typesetting:-msemantics = \"in ert\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\" , accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typeset ting:-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"i talic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"f alse\", separator = \"false\", stretchy = \"false\", symmetric = \"fal se\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:- mrow(Typesetting:-mo(\"&uminus0;\", mathvariant = \"normal\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesettin g:-mn(\"1\", mathvariant = \"normal\")), Typesetting:-mi(\"\")), Types etting:-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \+ \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \+ \"false\", separator = \"false\", stretchy = \"false\", symmetric = \" false\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesettin g:-mn(\"2\", mathvariant = \"normal\")), superscriptshift = \"2\", sub scriptshift = \"0\"), Typesetting:-mi(\"\"), Typesetting:-mrow(Typeset ting:-msubsup(Typesetting:-mo(\"∫\", foreground = \"[144,144,144] \", mathvariant = \"normal\", Typesetting:-msemantics = \"inert\", fen ce = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mrow (Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), \+ Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", se parator = \"false\", stretchy = \"false\", symmetric = \"false\", larg eop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mrow(Types etting:-msup(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \+ \"italic\"), Typesetting:-mn(\"2\", mathvariant = \"normal\"), supersc riptshift = \"0\")), Typesetting:-mi(\"\")), Typesetting:-mrow(Typeset ting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesett ing:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator \+ = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \" false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2 777778em\", rspace = \"0.2777778em\"), Typesetting:-mrow(Typesetting:- mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-m o(\"+\", mathvariant = \"normal\", fence = \"false\", separator = \"fa lse\", stretchy = \"false\", symmetric = \"false\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", lspace = \"0.222222 2em\", rspace = \"0.2222222em\"), Typesetting:-mn(\"2\", mathvariant = \"normal\")), Typesetting:-mi(\"\")), superscriptshift = \"2\", subsc riptshift = \"0\"), Typesetting:-mfenced(Typesetting:-mrow(Typesetting :-mn(\"4\", mathvariant = \"normal\"), Typesetting:-mo(\"−\", ma thvariant = \"normal\", fence = \"false\", separator = \"false\", stre tchy = \"false\", symmetric = \"false\", largeop = \"false\", movablel imits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspac e = \"0.2222222em\"), Typesetting:-mi(\"x\", italic = \"true\", mathva riant = \"italic\")), mathvariant = \"normal\"), Typesetting:-mi(\"\") , Typesetting:-mspace(height = \"0.0ex\", width = \"0.3em\", depth = \+ \"0.0ex\", linebreak = \"auto\"), Typesetting:-mo(\"ⅆ\", foreground = \"[144,144,144]\", mathvariant = \"normal\", Typesetting :-msemantics = \"inert\", fence = \"false\", separator = \"false\", st retchy = \"false\", symmetric = \"false\", largeop = \"false\", movabl elimits = \"false\", accent = \"false\", lspace = \"0.0em\", rspace = \+ \"0.0em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \" italic\")), Typesetting:-mi(\"\"), Typesetting:-mspace(height = \"0.0e x\", width = \"0.3em\", depth = \"0.0ex\", linebreak = \"auto\"), Type setting:-mo(\"ⅆ\", foreground = \"[144,144,144]\", mathv ariant = \"normal\", Typesetting:-msemantics = \"inert\", fence = \"fa lse\", separator = \"false\", stretchy = \"false\", symmetric = \"fals e\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", lspace = \"0.0em\", rspace = \"0.0em\"), Typesetting:-mi(\"y\", it alic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F' -F#6)-I(msubsupGF$6'-I#moGF$6/Q&∫F'/%+foregroundGQ.[144,144,144]F' /%,mathvariantGQ'normalF'/I+msemanticsGF$Q&inertF'/%&fenceGQ&falseF'/% *separatorGFC/%)stretchyGFC/%*symmetricGFC/%(largeopGFC/%.movablelimit sGFC/%'accentGFC/%'lspaceGQ&0.0emF'/%'rspaceGFR-F#6&-F,6%Q\"yF'/%'ital icGQ%trueF'/FFAFDFFF HFJFLFNFPFSF[qF+FfrFdsFWF+" }{TEXT 204 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT 247 33 "Now let \222s set this up in Maple.\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "C:=Doubleint(4-x,x=y^2..(y+2),y=-1..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(C);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 247 0 "" }}{PARA 0 "" 0 "" {TEXT 242 14 "C3M8 problems:" }{TEXT 247 98 " The boundaries for solids in the first octant are given. Using \+ Maple, sketch the region in the " }{TEXT 243 2 "xy" }{TEXT 247 202 "-p lane that is the base for the solid. Set up a double (iterated) integ ral whose value is the volume of the solid. Use Maple to evaluate the integral. Sketch the solid, either by hand or with Maple.\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT 247 5 "1. " }{XPPEDIT 18 0 "Typesetting :-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", \+ mathvariant = \"normal\", fence = \"false\", separator = \"false\", st retchy = \"false\", symmetric = \"false\", largeop = \"false\", movabl elimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rsp ace = \"0.2777778em\"), Typesetting:-mrow(Typesetting:-msup(Typesettin g:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting :-mn(\"2\", mathvariant = \"normal\"), superscriptshift = \"0\")), Typ esetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulename G6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6&-F,6%Q\"yF'/%'itali cGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fen ceGQ&falseF'/%*separatorGFB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/ %.movablelimitsGFB/%'accentGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F# 6#-I%msupGF$6%-F,6%Q\"xF'F4F7-I#mnGF$6$Q\"2F'F>/%1superscriptshiftGQ\" 0F'F+F+" }{TEXT 247 4 " , " }{XPPEDIT 18 0 "Typesetting:-mrow(Typeset ting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:- mrow(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic \"), Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \"false \", separator = \"false\", stretchy = \"false\", symmetric = \"false\" , largeop = \"false\", movablelimits = \"false\", accent = \"false\", \+ lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mi( \"y\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mo( \"=\", mathvariant = \"normal\", fence = \"false\", separator = \"fals e\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em \", rspace = \"0.2777778em\"), Typesetting:-mn(\"2\", mathvariant = \" normal\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typ esettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6&F+-F#6%-F,6%Q\"xF'/%'italicG Q%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q\"+F'/F:Q'normalF'/%&fence GQ&falseF'/%*separatorGFD/%)stretchyGFD/%*symmetricGFD/%(largeopGFD/%. movablelimitsGFD/%'accentGFD/%'lspaceGQ,0.2222222emF'/%'rspaceGFS-F,6% Q\"yF'F6F9-F=6-Q\"=F'F@FBFEFGFIFKFMFO/FRQ,0.2777778emF'/FUFgn-I#mnGF$6 $Q\"2F'F@F+" }{TEXT 247 3 " , " }{XPPEDIT 18 0 "Typesetting:-mrow(Type setting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"y\", italic = \+ \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvarian t = \"normal\", fence = \"false\", separator = \"false\", stretchy = \+ \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0 .2777778em\"), Typesetting:-mn(\"0\", mathvariant = \"normal\")), Type setting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sysl ibGF'6%-I#miGF$6#Q!F'-F#6%-F,6%Q\"yF'/%'italicGQ%trueF'/%,mathvariantG Q'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorG FB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'acce ntGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-I#mnGF$6$Q\"0F'F>F+" }{TEXT 247 3 " , " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), \+ Typesetting:-mrow(Typesetting:-mi(\"z\", italic = \"true\", mathvarian t = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fenc e = \"false\", separator = \"false\", stretchy = \"false\", symmetric \+ = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typese tting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \" italic\"), Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \" false\", separator = \"false\", stretchy = \"false\", symmetric = \"fa lse\", largeop = \"false\", movablelimits = \"false\", accent = \"fals e\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting: -mn(\"2\", mathvariant = \"normal\")), Typesetting:-mi(\"\")), Typeset ting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibG F'6%-I#miGF$6#Q!F'-F#6&-F,6%Q\"zF'/%'italicGQ%trueF'/%,mathvariantGQ'i talicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/ %)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'accentG FB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F#6%-F,6%Q\"xF'F4F7-F;6-Q\"+F' F>F@FCFEFGFIFKFM/FPQ,0.2222222emF'/FSFgn-I#mnGF$6$Q\"2F'F>F+F+" }{TEXT 247 3 " , " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), \+ Typesetting:-mrow(Typesetting:-mi(\"z\", italic = \"true\", mathvarian t = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fenc e = \"false\", separator = \"false\", stretchy = \"false\", symmetric \+ = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typese tting:-mn(\"0\", mathvariant = \"normal\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F' -F#6%-F,6%Q\"zF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q \"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/%)stretchyGFB/%*sy mmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB/%'lspaceGQ,0.27 77778emF'/%'rspaceGFQ-I#mnGF$6$Q\"0F'F>F+" }{TEXT 247 6 " " } {TEXT 244 7 "BEWARE!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 247 5 "2. " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\" ), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", lsp ace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mrow(Ty pesetting:-mn(\"2\", mathvariant = \"normal\"), Typesetting:-mo(\"&Inv isibleTimes;\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \+ \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0 .0em\", rspace = \"0.0em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\")), Typesetting:-mi( \"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#mi GF$6#Q!F'-F#6&-F,6%Q\"yF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I #moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/%)stretch yGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB/%'lspa ceGQ,0.2777778emF'/%'rspaceGFQ-F#6%-I#mnGF$6$Q\"2F'F>-F;6-Q1&Invisible Times;F'F>F@FCFEFGFIFKFM/FPQ&0.0emF'/FSFhn-F,6%Q\"xF'F4F7F+F+" }{TEXT 247 4 " , " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), \+ Typesetting:-mrow(Typesetting:-mi(\"y\", italic = \"true\", mathvarian t = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fenc e = \"false\", separator = \"false\", stretchy = \"false\", symmetric \+ = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), Typese tting:-mrow(Typesetting:-msup(Typesetting:-mi(\"x\", italic = \"true\" , mathvariant = \"italic\"), Typesetting:-mn(\"2\", mathvariant = \"no rmal\"), superscriptshift = \"0\")), Typesetting:-mi(\"\")), Typesetti ng:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF' 6%-I#miGF$6#Q!F'-F#6&-F,6%Q\"yF'/%'italicGQ%trueF'/%,mathvariantGQ'ita licF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/%) stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB /%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F#6#-I%msupGF$6%-F,6%Q\"xF'F4F7- I#mnGF$6$Q\"2F'F>/%1superscriptshiftGQ\"0F'F+F+" }{TEXT 247 4 " , " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"z\", italic = \"true\", mathvariant = \"italic\" ), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", lsp ace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mrow(Ty pesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typ esetting:-mo(\"+\", mathvariant = \"normal\", fence = \"false\", separ ator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mi(\"y\", ita lic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\")), Ty pesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sy slibGF'6%-I#miGF$6#Q!F'-F#6&-F,6%Q\"zF'/%'italicGQ%trueF'/%,mathvarian tGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separato rGFB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'ac centGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F#6%-F,6%Q\"xF'F4F7-F;6-Q \"+F'F>F@FCFEFGFIFKFM/FPQ,0.2222222emF'/FSFgn-F,6%Q\"yF'F4F7F+F+" } {TEXT 247 4 " , " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi( \"\"), Typesetting:-mrow(Typesetting:-mi(\"z\", italic = \"true\", mat hvariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\") , Typesetting:-mn(\"0\", mathvariant = \"normal\")), Typesetting:-mi( \"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#mi GF$6#Q!F'-F#6%-F,6%Q\"zF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I #moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/%)stretch yGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB/%'lspa ceGQ,0.2777778emF'/%'rspaceGFQ-I#mnGF$6$Q\"0F'F>F+" }{TEXT 247 1 " " } }}{EXCHG {PARA 0 "" 0 "" {TEXT 247 4 "3. " }{XPPEDIT 18 0 "Typesettin g:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"y\" , italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", s tretchy = \"false\", symmetric = \"false\", largeop = \"false\", movab lelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rs pace = \"0.2777778em\"), Typesetting:-mi(\"x\", italic = \"true\", mat hvariant = \"italic\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+module nameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%-F,6%Q\"yF'/%'i talicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/% &fenceGQ&falseF'/%*separatorGFB/%)stretchyGFB/%*symmetricGFB/%(largeop GFB/%.movablelimitsGFB/%'accentGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGF Q-F,6%Q\"xF'F4F7F+" }{TEXT 247 3 " , " }{XPPEDIT 18 0 "Typesetting:-mr ow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typ esetting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"+\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \+ \"false\", largeop = \"false\", movablelimits = \"false\", accent = \" false\", lspace = \"0.2222222em\", rspace = \"0.2222222em\"), Typesett ing:-mi(\"y\", italic = \"true\", mathvariant = \"italic\")), Typesett ing:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator \+ = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \" false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2 777778em\", rspace = \"0.2777778em\"), Typesetting:-mn(\"2\", mathvari ant = \"normal\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG 6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6&F+-F#6%-F,6%Q\"xF'/% 'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q\"+F'/F:Q'normalF' /%&fenceGQ&falseF'/%*separatorGFD/%)stretchyGFD/%*symmetricGFD/%(large opGFD/%.movablelimitsGFD/%'accentGFD/%'lspaceGQ,0.2222222emF'/%'rspace GFS-F,6%Q\"yF'F6F9-F=6-Q\"=F'F@FBFEFGFIFKFMFO/FRQ,0.2777778emF'/FUFgn- I#mnGF$6$Q\"2F'F@F+" }{TEXT 247 4 " , " }{XPPEDIT 18 0 "Typesetting:- mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"x\", i talic = \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", ma thvariant = \"normal\", fence = \"false\", separator = \"false\", stre tchy = \"false\", symmetric = \"false\", largeop = \"false\", movablel imits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspac e = \"0.2777778em\"), Typesetting:-mn(\"2\", mathvariant = \"normal\") ), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingG I(_syslibGF'6%-I#miGF$6#Q!F'-F#6%-F,6%Q\"xF'/%'italicGQ%trueF'/%,mathv ariantGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*sep aratorGFB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB /%'accentGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-I#mnGF$6$Q\"2F'F>F+" }{TEXT 247 3 " , " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi( \"\"), Typesetting:-mrow(Typesetting:-mi(\"z\", italic = \"true\", mat hvariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal \", fence = \"false\", separator = \"false\", stretchy = \"false\", sy mmetric = \"false\", largeop = \"false\", movablelimits = \"false\", a ccent = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\") , Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\")) , Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI (_syslibGF'6%-I#miGF$6#Q!F'-F#6%-F,6%Q\"zF'/%'italicGQ%trueF'/%,mathva riantGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*sepa ratorGFB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/ %'accentGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F,6%Q\"yF'F4F7F+" } {TEXT 247 3 " , " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\" \"), Typesetting:-mrow(Typesetting:-mi(\"z\", italic = \"true\", mathv ariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symme tric = \"false\", largeop = \"false\", movablelimits = \"false\", acce nt = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), T ypesetting:-mn(\"0\", mathvariant = \"normal\")), Typesetting:-mi(\"\" ));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6 #Q!F'-F#6%-F,6%Q\"zF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moG F$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/%)stretchyGFB /%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB/%'lspaceGQ ,0.2777778emF'/%'rspaceGFQ-I#mnGF$6$Q\"0F'F>F+" }{TEXT 247 5 " " } {TEXT 245 7 "BEWARE!" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 247 9 "4. Let " }{TEXT 246 1 "k" }{TEXT 247 67 " be an instructor assigned constant as described in the preface. " }{XPPEDIT 18 0 "Typesetting:-mrow(Typ esetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"y\", italic = \+ \"true\", mathvariant = \"italic\"), Typesetting:-mo(\"=\", mathvarian t = \"normal\", fence = \"false\", separator = \"false\", stretchy = \+ \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2777778em\", rspace = \"0 .2777778em\"), Typesetting:-mrow(Typesetting:-mi(\"k\", italic = \"tru e\", mathvariant = \"italic\"), Typesetting:-mo(\"−\", mathvaria nt = \"normal\", fence = \"false\", separator = \"false\", stretchy = \+ \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \"0.2222222em\", rspace = \"0 .2222222em\"), Typesetting:-mi(\"x\", italic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mr owG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6&- F,6%Q\"yF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q\"=F'/ F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/%)stretchyGFB/%*symmetri cGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB/%'lspaceGQ,0.2777778e mF'/%'rspaceGFQ-F#6%-F,6%Q\"kF'F4F7-F;6-Q(−F'F>F@FCFEFGFIFKFM/FP Q,0.2222222emF'/FSFgn-F,6%Q\"xF'F4F7F+F+" }{TEXT 247 3 " , " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"y\", italic = \"true\", mathvariant = \"italic\" ), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", lsp ace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mrow(Ty pesetting:-mi(\"k\", italic = \"true\", mathvariant = \"italic\"), Typ esetting:-mo(\"+\", mathvariant = \"normal\", fence = \"false\", separ ator = \"false\", stretchy = \"false\", symmetric = \"false\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", lspace = \+ \"0.2222222em\", rspace = \"0.2222222em\"), Typesetting:-mi(\"x\", ita lic = \"true\", mathvariant = \"italic\")), Typesetting:-mi(\"\")), Ty pesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_sy slibGF'6%-I#miGF$6#Q!F'-F#6&-F,6%Q\"yF'/%'italicGQ%trueF'/%,mathvarian tGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separato rGFB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'ac centGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-F#6%-F,6%Q\"kF'F4F7-F;6-Q \"+F'F>F@FCFEFGFIFKFM/FPQ,0.2222222emF'/FSFgn-F,6%Q\"xF'F4F7F+F+" } {TEXT 247 3 " , " }{XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\" \"), Typesetting:-mrow(Typesetting:-mi(\"x\", italic = \"true\", mathv ariant = \"italic\"), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symme tric = \"false\", largeop = \"false\", movablelimits = \"false\", acce nt = \"false\", lspace = \"0.2777778em\", rspace = \"0.2777778em\"), T ypesetting:-mn(\"1\", mathvariant = \"normal\")), Typesetting:-mi(\"\" ));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6 #Q!F'-F#6%-F,6%Q\"xF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moG F$6-Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/%)stretchyGFB /%*symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB/%'lspaceGQ ,0.2777778emF'/%'rspaceGFQ-I#mnGF$6$Q\"1F'F>F+" }{TEXT 247 3 " , " } {XPPEDIT 18 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-m row(Typesetting:-mi(\"z\", italic = \"true\", mathvariant = \"italic\" ), Typesetting:-mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \"false\", stretchy = \"false\", symmetric = \"false\", l argeop = \"false\", movablelimits = \"false\", accent = \"false\", lsp ace = \"0.2777778em\", rspace = \"0.2777778em\"), Typesetting:-mrow(Ty pesetting:-msup(Typesetting:-mi(\"x\", italic = \"true\", mathvariant \+ = \"italic\"), Typesetting:-mn(\"2\", mathvariant = \"normal\"), super scriptshift = \"0\")), Typesetting:-mi(\"\")), Typesetting:-mi(\"\")); " "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6#Q! F'-F#6&-F,6%Q\"zF'/%'italicGQ%trueF'/%,mathvariantGQ'italicF'-I#moGF$6 -Q\"=F'/F8Q'normalF'/%&fenceGQ&falseF'/%*separatorGFB/%)stretchyGFB/%* symmetricGFB/%(largeopGFB/%.movablelimitsGFB/%'accentGFB/%'lspaceGQ,0. 2777778emF'/%'rspaceGFQ-F#6#-I%msupGF$6%-F,6%Q\"xF'F4F7-I#mnGF$6$Q\"2F 'F>/%1superscriptshiftGQ\"0F'F+F+" }{TEXT 247 3 " , " }{XPPEDIT 18 0 " Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting :-mi(\"z\", italic = \"true\", mathvariant = \"italic\"), Typesetting: -mo(\"=\", mathvariant = \"normal\", fence = \"false\", separator = \" false\", stretchy = \"false\", symmetric = \"false\", largeop = \"fals e\", movablelimits = \"false\", accent = \"false\", lspace = \"0.27777 78em\", rspace = \"0.2777778em\"), Typesetting:-mn(\"0\", mathvariant \+ = \"normal\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I ,TypesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6%-F,6%Q\"zF'/%'italicGQ%t rueF'/%,mathvariantGQ'italicF'-I#moGF$6-Q\"=F'/F8Q'normalF'/%&fenceGQ& falseF'/%*separatorGFB/%)stretchyGFB/%*symmetricGFB/%(largeopGFB/%.mov ablelimitsGFB/%'accentGFB/%'lspaceGQ,0.2777778emF'/%'rspaceGFQ-I#mnGF$ 6$Q\"0F'F>F+" }{TEXT 247 3 " . " }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 15 10 1804 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }