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It can also quickly compute multiple integrals. We de monstrate commands for performing some of these operations in the foll owing discussion. We begin by clearing all variables" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 20 "Plotting 3D surfaces" }}{PARA 0 "" 0 "" {TEXT -1 116 "Maple can easily plot surfaces in 3D space. We demonstrate with the following examples. We start by entering in the \+ " }{TEXT 258 6 "plots " }{TEXT -1 8 "package." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "War ning, the name changecoords has been redefined\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Examp le 1" }}{PARA 0 "" 0 "" {TEXT -1 37 "Suppose we want to plot the graph of " }{XPPEDIT 18 0 "f(x,y) = exp(-x^2-y^2);" "6#/-%\"fG6$%\"xG%\"yG- %$expG6#,&*$F'\"\"#!\"\"*$F(F.F/" }{TEXT -1 41 ". We enter this as a f unction as follows:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "f := \+ (x, y) -> exp(-x^2-y^2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6$ %\"xG%\"yG6\"6$%)operatorG%&arrowGF)-%$expG6#,&*$)9$\"\"#\"\"\"!\"\"*$ )9%F4F5F6F)F)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "Using the func tion -> arrow notation allows one to easily evaluate " }{TEXT 259 1 "f " }{TEXT -1 57 " at specific values. For example, this is the result f or " }{TEXT 260 2 "f " }{TEXT -1 6 "(1, 2)" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 8 "f(1, 2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$expG6 #!\"&" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "To see what this surface looks like in 3D, the " }{TEXT 261 7 "plot3d " }{TEXT -1 52 "commmand is useful. The following command will plot " }{TEXT 262 1 "z" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "f(x,y) = exp(-x^2-y^2)" "6#/-%\"fG6$%\"xG% \"yG-%$expG6#,&*$F'\"\"#!\"\"*$F(F.F/" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 107 "plot3d(f(x,y), x = -5..5, y = -5.. 5, color = blue, axes = NORMAL, title = \"Plot of f(x,y) = e^(-x^2-y^2 )\");" }}{PARA 13 "" 1 "" {GLPLOT3D 476 476 476 {PLOTDATA 3 "6&-%%GRID G6&;$!\"&\"\"!$\"\"&F)F&X,%)anythingG6\"6\"[gl'!%\"!!#\\bm\":\":3B6D25 7D547E08403BC8B15119C5D5543C1D90BA45441EFD3C6903DAEC8F0FB13CADE99CFC94 6B343CE946A9414E102F3D1E2F77B039ECA13D497938FC7578053D6E61C0D5E647E73D 899B264809E0B33D9E8014E623D6AB3DA9AC2DDBE4B31F3DAE8A37A45FC32F3DA9AC2D DBE4B31F3D9E8014E623D6AB3D899B264809E0B33D6E61C0D5E647E73D497938FC7578 053D1E2F77B039ECA13CE946A9414E102F3CADE99CFC946B343C6903DAEC8F0FB13C1D 90BA45441EFD3BC8B15119C5D5543B6D257D547E08403BC8B15119C5D5543C24EB5F04 8BDA823C790C2B058502903CC5314C0AD0BA123D095778A486F1303D4569E5018E56D4 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allows you to turn it. In the previous command, the first paramet er specifies the function to be plotted (which is stored as " }{TEXT 263 1 "f" }{TEXT -1 2 " (" }{TEXT 264 4 "x, y" }{TEXT -1 48 ")), the s econd and third are the ranges for the " }{TEXT 265 1 "x" }{TEXT -1 5 " and " }{TEXT 266 1 "y" }{TEXT -1 186 " axes, the fourth specifies th at the color is blue, the fifth specifies the axes type (it can be NOR MAL, FRAMED, BOXED, or NONE), and the sixth parameter specifies a titl e for the graph." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 82 "Sometimes we many want a better graph solution. By defaul t, Maple will divide the " }{TEXT 267 3 "x-y" }{TEXT -1 63 " axes grid up into 625 (25 x 25) points, form the order pairs (" }{TEXT 293 4 "x , y" }{TEXT -1 118 ") from this grid, and use these to find the z coo rdinate. If we want more points for a better resolution, we use the " }{TEXT 268 9 "numpoints" }{TEXT -1 101 " option. The following command will increase the number of points plotted to 1000 with a FRAMED axes ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "plot3d(f(x,y), x = -5 ..5, y = -5..5, color = blue, axes = FRAMED, numpoints = 2000, title = \"Plot of f(x,y) = e^(-x^2-y^2)\");" }}{PARA 13 "" 1 "" {GLPLOT3D 476 476 476 {PLOTDATA 3 "6&-%%GRIDG6&;$!\"&\"\"!$\"\"&F)F&X,%)anything G6\"6\"[gl'!%\"!!#__s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66D-%&COLORG6&%$RGBG$F)F)F4$\"*++++\"!\")-%&TITLEG6#Q>Plot~of~f(x,y)~= ~e^(-x^2-y^2)F/-%*AXESSTYLEG6#%&FRAMEG-%+AXESLABELSG6%%\"xG%\"yGQ!F/" 1 2 0 1 10 0 2 1 1 3 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Maple also has a nice command, " } {TEXT 269 11 "contourplot" }{TEXT -1 25 ", which is a part of the " } {TEXT 270 5 "plots" }{TEXT -1 97 " package, for doing 2D contour plot s. 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computed and plotted. We will need the " }{TEXT 271 5 "pl ots" }{TEXT -1 9 " package:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots);" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name cha ngecoords has been redefined\n" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7gn% ,InteractiveG%(animateG%*animate3dG%-animatecurveG%&arrowG%-changecoor dsG%,complexplotG%.complexplot3dG%*conformalG%,conformal3dG%,contourpl otG%.contourplot3dG%*coordplotG%,coordplot3dG%-cylinderplotG%,densityp lotG%(displayG%*display3dG%*fieldplotG%,fieldplot3dG%)gradplotG%+gradp lot3dG%,graphplot3dG%-implicitplotG%/implicitplot3dG%(inequalG%,intera ctiveG%2interactiveparamsG%-listcontplotG%/listcontplot3dG%0listdensit yplotG%)listplotG%+listplot3dG%+loglogplotG%(logplotG%+matrixplotG%)mu ltipleG%(odeplotG%'paretoG%,plotcompareG%*pointplotG%,pointplot3dG%*po larplotG%,polygonplotG%.polygonplot3dG%4polyhedra_supportedG%.polyhedr aplotG%'replotG%*rootlocusG%,semilogplotG%+setoptionsG%-setoptions3dG% +spacecurveG%1sparsematrixplotG%+sphereplotG%)surfdataG%)textplotG%+te xtplot3dG%)tubeplotG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Suppose w e considered the surface given by the ellipsoid " }{XPPEDIT 18 0 "9*x^ 2+4*y^2+z^2 = 36;" "6#/,(*&\"\"*\"\"\"*$%\"xG\"\"#F'F'*&\"\"%F'*$%\"yG F*F'F'*$%\"zGF*F'\"#O" }{TEXT -1 178 ". We need to form the general eq uation by getting all terms on one side of the equation of the surface set equal to zero. Hence, we enter the surface with the following equ ation:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "F := 9*x^2 + 4*y^ 2 + z^2 - 36;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,**&\"\"*\"\"\" )%\"xG\"\"#F(F(*&\"\"%F()%\"yGF+F(F(*$)%\"zGF+F(F(\"#O!\"\"" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Suppose we want to find the equati on of the tangent plane at the point (1, 2, " }{XPPEDIT 18 0 "sqrt(11 );" "6#-%%sqrtG6#\"#6" }{TEXT -1 172 "). To find the equation of any p lane, including a tangent plane, we need a point and a normal vector . The point is already given. Recall that the gradient of F, given by \+ " }{XPPEDIT 18 0 "F[x]*i+F[y]*j+F[z]*k;" "6#,(*&&%\"FG6#%\"xG\"\"\"%\" iGF)F)*&&F&6#%\"yGF)%\"jGF)F)*&&F&6#%\"zGF)%\"kGF)F)" }{TEXT -1 252 ", is a vector normal to the surface and hence normal to the tangent pla ne. This will be the normal vector we use. The next 3 commands compute the components of the gradient vector using partial derivatives and m ake the results into functions using the " }{TEXT 272 7 "unapply" } {TEXT -1 10 " function." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 " Fx := unapply(diff(F, x), x, y, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#FxGf*6%%\"xG%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,$*&\"#=\"\"\"9$F1 F1F*F*F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Fy := unapply(d iff(F, y), x, y, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FyGf*6%%\"x G%\"yG%\"zG6\"6$%)operatorG%&arrowGF*,$*&\"\")\"\"\"9%F1F1F*F*F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Fz := unapply(diff(F, z), x, y, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#FzGf*6%%\"xG%\"yG%\"zG6 \"6$%)operatorG%&arrowGF*,$*&\"\"#\"\"\"9&F1F1F*F*F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "The next statements enters the the coordinates \+ of the given point." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "x0 : = 1; y0 := 2; z0 := sqrt(11);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%# x0G\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#y0G\"\"#" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#>%#z0G*$\"#6#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "By using the basic standard equation for a plane g, \+ we enter and store the tangent plane equation:" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 90 "tplane := Fx(x0, y0, z0)*(x - x0) + Fy(x0, y0, z0)*(y - y0) + Fz(x0, y0, z0)*(z - z0) = 0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'tplaneG/,**&\"#=\"\"\"%\"xGF)F)\"#]!\"\"*&\"#;F)%\"y GF)F)*(\"\"#F)\"#6#F)F1,&%\"zGF)*$F2F3F,F)F)\"\"!" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 84 "To plot the tangent plane, it will be easier to so lve it in terms of the variable z." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "tplane := solve(tplane, z);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'tplaneG,$*(\"#6!\"\",(*&\"\"*\"\"\"%\"xGF,F,\"#OF(*& \"\")F,%\"yGF,F,F,F'#F,\"\"#F(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 209 "Since the gradient vector is tangent to the surface, it will be t he parallel vector we need to write the equation of the normal line. T he following command computes the parametric equations of the normal l ine:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "xp := x0 + Fx(x0, y 0, z0)*t; yp := y0 + Fy(x0, y0, z0)*t; zp := z0 + Fz(x0, y0, z0)*t; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#xpG,&\"\"\"F&*&\"#=F&%\"tGF&F& " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#ypG,&\"\"#\"\"\"*&\"#;F'%\"tGF' F'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#zpG,&*$\"#6#\"\"\"\"\"#F)*(F* F)F'F(%\"tGF)F)" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 87 "We now plot th e previous results. To set up the plot of the ellipsoid, we will use t he " }{TEXT 273 14 "implicitplot3d" }{TEXT -1 77 " command, which is d esigned to plot implicit equations involving 3 variables:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "p1 := implicitplot3d(F = 0, x = -4 ..4, y = -4..4, z = -7..7, numpoints = 1000, color = blue, axes = fram ed): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "The next command sets up the tangent plane plot using " }{TEXT 274 6 "plot3d" }{TEXT -1 1 ":" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "p2 := plot3d(tplane, x = \+ 0..3, y = 0..3, color = red):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 " We next set up the normal line plot using the " }{TEXT 275 10 "spacecu rve" }{TEXT -1 107 " command. Once you graph this line on the same gra ph with the surfaces, you may have to vary the values of " }{TEXT 276 1 "t" }{TEXT -1 26 " to obtain a decent graph." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 76 "p3 := spacecurve([xp, yp, zp], t = -0.5..0.5, \+ color = black, thickness = 2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "We next display our results on the same graph. 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COLORG6&FcalFablF-F--%*AXESSTYLEG6#%&FRAMEG-%+AXESLABELSG6&Q\"xF^blQ\" yF^blQ\"zF^bl-%%FONTG6#%(DEFAULTG" 1 2 0 1 10 0 2 1 1 3 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 37 "Computing Double and \+ Triple Integrals" }}{PARA 0 "" 0 "" {TEXT -1 181 "Maple has the abilit y to quickly compute multiple integrals. We will demonstrate this for \+ double and triple integrals. The package will that contains the routin es we need is in the " }{TEXT 277 7 "student" }{TEXT -1 72 " package. \+ We read this into our session and see what is available, enter" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student);" }}{PARA 12 " " 1 "" {XPPMATH 20 "6#7@%\"DG%%DiffG%*DoubleintG%$IntG%&LimitG%(Linein tG%(ProductG%$SumG%*TripleintG%*changevarG%/completesquareG%)distanceG %'equateG%*integrandG%*interceptG%)intpartsG%(leftboxG%(leftsumG%)make procG%*middleboxG%*middlesumG%)midpointG%(powsubsG%)rightboxG%)rightsu mG%,showtangentG%(simpsonG%&slopeG%(summandG%*trapezoidG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 1" }}{PARA 0 "" 0 "" {TEXT -1 47 "Suppose we want to com pute the double integral " }{OLE 1 4609 1 "[xm]Br=WfoRrB:::wk;nyyI;G:; :j::>:B>N:F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::fyyyyya:nYf::G:I:wAyA::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:EL q^H_mvJ::::::::gjpbH\\:Oqqwr;V:>j@Z:j:vCSmlJ ::::::::::OJ;@jyyyyyI:;Z:::::::^<>:F:AlqfG[maNFO=;::::::::_J;@j:j<>:ya yA:<::::::=J:nF>:V:Y:G:;:wA?:Jy yyyg:n:nY>;F;N;;j?>:S:UJ:n;v;;JBB:]:_J:V<^a; ^[;^^Hq[:^^Hq;;J::YbfYYm[bmW:;j:B:CTQsyKB:=Jyyy;d:yayQZHQ:>:;`:Z@w:Vx:Xx:Xv:L:V:CJ>@:@<:F\\;OtX?<b:K::>v:?<<:GJ HJ;vd^qp=@j;\\;DJ;]\\LVjrs:YZEF\\;NZBP@N[ klb>A^LRXDJ=i\\LVJHjA:J:J;a\\AFSDk;aa<:L@\\a:V[;NZB@@@JC HRvV:^[KFZA=U:B:qIZqIZyyyxy;B:CJ:FZ:>Z:vYxY;<:;:Y H>:EZ:F[::::::;C:?jysy: >:;B::::::^::<::::: :::j=\\:B :;xyyQWyyyyY:x:oR;V[:JMJ@fc[_hb_ds?h_KB@JHB:qi:;XyB:^qaTXUeRYE U>qDN[JAglQGufC:US:>;;JSdJ\\IJZFZ:>Z:Nxs?;N@>g;F :NXsZ:VY;fyB:^Qb::vG>dQN[e@glQF:f_s_=:CB:>X=B:K :_Ks@j`FZ:B:]SXK:_KGPJbF:>`s?;N@>i;F:;JR:_KDEj^dj:Zs?;bZ:JB[:^Z<::::=@;;:f?=JZ:RXjw?JB:>LB:C::::FR>>:jP>:C B:>X=J>JSdJZFZEF:JSjZQj:<:`Q>ZDjaDj:>:KSXs:qQjuI<:[n:>:Cb:^D:::= @;;:f?=JZ:>is_=LJ<::::=@;;:f?;J<<:[q:>;N@;lAq J;Jl\\Q>JSj`iJ;Jl`Q>JSj:aj:>Z:Fis?;N@Fm@N:;Jv>;N@Vw>:jPF:C:[Q:>;N@f`GF:JSjshj:JJN;:s ?OV:B:J;B: ::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::2:" }{TEXT -1 12 ". Using the " }{TEXT 278 9 "Doubleint" }{TEXT -1 33 " co mmand, we set this as follows:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "Doubleint(x^2 - 2*x*y + y^2, y = 0..x^2, x = 0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$,(*$)%\"xG\"\"#\"\"\"F-*(F,F-F+F- %\"yGF-!\"\"*$)F/F,F-F-/F/;\"\"!F)/F+;F5F-" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 247 "In the previous command, the first parameter is the func tion you are integrating. The limits of integration are entered from f rom the inside out in the order that the integral will be computed. Fi rst, the inside limts with respect to y ( y = 0 .. " }{XPPEDIT 18 0 "x ^2;" "6#*$%\"xG\"\"#" }{TEXT -1 68 ") are entered. Then the outside li mits for x (x = 0..1) are entered." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 4 "The " }{TEXT 279 9 "Doubleint" }{TEXT -1 1 " " }{TEXT 280 0 "" }{TEXT -1 88 "only sets up the double integral, \+ but does not display the result. To evaluate, use the " }{TEXT 281 5 " value" }{TEXT -1 34 " function on the previous command." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6##\"#<\"$5#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 2" }}{PARA 0 "" 0 " " {TEXT -1 201 "An interesting fact to note is that Maple has the abil ity to compute double integrals without a necessary switch of the limi ts for integration. As an example, suppose we to compute the double in tegral " }{OLE 1 4609 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F:nyyyyy ]::yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::fyyyyya:nYf::G:I:wAyA::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::::::::::::::::::::::::NDYmq^H;C:ELq^H_mvJ::::::::gj< vyyuyJ;B:?bm?:;JZC:bKB::>pbH\\:Oqqwr;V:>j@Z:j:vCSmlJ::::::::::OJ;@jyyy yyI:;Z:::::::^<>:F:AlqfG[maNFO=;::::::::_J;@j:j<>:yayA:<::::::=J:nF>:V :Y:G:;:wA?:Jyyyyg:n:nY>;F;N;;j? >:S:UJ:n;v;;JBB:]:_J:V<^Z:V:CJJf@SKJ vBEMqfHMMqfFMDvCA;:;TggJlqOEB:;j;@j:HJ?dj: HJFR:AB@b:?FDDk;Q`=v;hjB@J;TRCDLCHRvVZIJ@@k:DJ>E=;B:dJ;DrNNZ;@>_rZCMCH RvNZ;E==J>A]:=B:;B:yayQZ:J:jCiZ:^w?JkL:=j>r:]c:N:s:;j>Wlj:g=;J::::::>^:N:yay=J:>Z::::::J@[C:> Z::::::::kJ;@j;>:C:yayA:<::::::I:CG;Z: f:^[<>f?>jDN\\:B:;xyyQOyyyyYZd:OR;V[:B:G;Sj`@Pt\\Pd`Qr P@[S:sJ:b:FZ:B:_YXK:_;yW:=:_QHB:qQjuY:B:^Qb::vG>`DN[@?glQl;Z:^Z>:jP@J:^Z:JrGZ:>;N@_S:Ul:JSjT@j:jR:_Kg_h^j::Uk :^:>X;J>JS>Z=Vc@F:;B:MSXK:_;Sr:=:KSXs:qQJxI;B:>LB:C:ZZ::jZKJ::UK:^Z:Jr G:K:_kcAJFQj:B:`QHjw?vx]:JBG:;J>:jPF:C:[Q:>;N`DNUMT:=J: JS^DOR;=:kUXsZ:VY;><:[>^::::jZKJ::UK:^Z:JrG:K:_K:VtBN:^fs?;N@v\\AN:;Jl \\Q>JSJBIj:^r?;N@fq@F: >>OZ:n>N;yayIZ:J:^=VY[j=B:;JX[:FZ:CB:>X=j; " 0 "" {MPLTEXT 1 0 46 "Doubleint(exp(-x^2), x = y/2 ..1/2, y = 0..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$-%$expG6#,$*$)%\"xG\" \"#\"\"\"!\"\"/F.;,$*&F/F1%\"yGF0F0#F0F//F6;\"\"!F0" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#,&\"\"\"F$-%$expG6##!\"\"\"\"%F)" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 9 "Example 3" }}{PARA 0 "" 0 "" {TEXT -1 29 "Suppose we want \+ to integrate " }{OLE 1 5121 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N:F: nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::fyyyyya:nYf::G:I:K:wAyA::::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;::: ::::n:;`:Z@[::JbxlC:<::::::JDJ:j:VBYmp>HYLkNG>::::::::N:;::::::::_Z:vyyuy:>:<::::::AJ:n:>:nYN::wyyyqZ :>;nYN;V;^;f;;JAjA>:[Z:F:i:k:m:o:q:s:u:w:y:;C:R<:TnEj``pkDqqHqqTPtWdZfbk;>ZEZ?GHZhV@cYH_WV>Z::j`>mKJVDcml>h :>H;B:AJ<>>eKP>>IllnGemnnGEm^JtJp>:[=@j;\\;Dj:=\\=>=:?RZ;;:JCHrlVZIj?@k:DJ>I]LVjrc:GvDDk; aa===J>A]<>kpb:Gf>BX:` a:f[Ob:=FBVj^c>AfXFJBSRXDj:[^[rC:Q:[;_rZkm;x:SR>=b:D]:>Z:f;J:h:G f>DJ>Am\\FHemj^HMmqnG;KaFFJufF;B:;:::::JJ:;B::::::^:;JB@Z:B: N:YLpfF[D:_mlVH[KR<:;B:::::::JFNZ;V:;JB:[Y;>Z:j::<::::::wqy[:::::::::::::vYxI:;Z:::::::::=b:KVF;B:U :B:;:jcFIUMkvGSM:>:::::yayY:^:FZ:>Z:vYxY;<:;:;D@D:;B:E:?R:=:E:=b:yyyyI:E:WS:kZ:f:V[f?b?_:Gc;YJGvyyuy;B: [B:^;UTRcETcTX[US>\\;^=;B:qi:;XyB:^qaT XUeRYEU>\\DN[[AglQ;N`D^r=vF=J:JSj AAj:Js;_;kD;=:_QHB:qQjuY:B:^Qb::vGv_XN[q@glQ:[Y:>;N`DN]? ^A=Z:f_s?;N@AU:YS;=:U;N@^v=F:f?JS^l\\DbrKev_:FR>>XA^:f_;>:C:[q:>;N`D>b=FgGF:l;B:q^lZKJ::Uk:^:>X;J>JSNh;^H=Z:F`s?;N@oj`GZ:jR:_;yV:=J: >`s?;N@AMYEJ;jRTM>JSN=KE;=:[cGK:_KDOtAF:;jSTMHB:qQJxI;B:>LB:C::::FR>>: jP>:CB:>X=J>JSFX]Y:=J:<:[>>:jPF:C:[Q:>;N`D>BYB>=B: JSjAuj:Zp>;b_p^=VY;iyB:>L=J:^::::jZKJ:Z:f?;J<<:[q:>;N@_mk @j:JSVFoY;=:meGK:_;?G;=:k=N@cnIhj:JwTM>JSjaaj:>:m=N@NhBF:>IsZ :VY;><:[>^Z<::::=@;;:f?=JJSj?EK;Jl`Q>JSjg=K;>:Ce XKJ:N@vfIF:<:_;aE==:meXKB:N@v]AF:fft_=VYZ:JX?::::FR>>:jP>:C B:>X=J>JSjsIk:B:;cXK:_;]c>=J:>>JSj:=k:JJN;;JSj_Pj;N@Nj>O B:n>N;yayIZ:J:^=VY[B:I:;JXK:OV:B:=B:^jnQX;j;J=:=:J::;N:B:>JAB:F::;:>J; Z:J:G:F::;:>J;Z:J:O:F::;:>J;Z:J:W:F::;:>J;Z:J:_Z:j::>:J:?:<:;n;j::>:J: ?:<:;N:J:?:<:;><:::::::::::::::::::::::::::::::::::4:" }{TEXT -1 6 ". The " }{TEXT 282 10 "Tripleint " }{TEXT -1 129 "command is useful for evaluating triple integrals. The following two commands sets up t he triple integral and evaluate its value." }}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 68 "Tripleint(exp(3*x+y-z)*(y+2*z), z = 0..16-x-y, y = \+ 0..x, x = -1..1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$-F$ 6$*&-%$expG6#,(*&\"\"$\"\"\"%\"xGF1F1%\"yGF1%\"zG!\"\"F1,&F3F1*&\"\"#F 1F4F1F1F1/F4;\"\"!,(\"#;F1F2F5F3F5/F3;F;F2/F2;F5F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #,2*&#\"\"\"\"\"$F&-%$expG6#!\"$F&F&*&#\"#P\"\")F&-F)6#!#?F&!\"\"*&#F& \"#;F&-F)6#!\"%F&F&*&#\"#>\"\"'F&-F)6#!#AF&F&*&#F&F'F&-F)6#F'F&F3*&#\" #LF/F&-F)6#!#7F&F&*&#\"\"(F6F&-F)6#\"\"%F&F&*&#F/F'F&-F)6#!#5F&F3" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "This is the decimal approximation \+ of the previous answer." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "e valf(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+Nc\"4s\"!\")" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 " " {TEXT -1 9 "Example 4" }}{PARA 0 "" 0 "" {TEXT -1 29 "Suppose we wan t to integrate " }{OLE 1 5121 1 "[xm]Br=WfoRrB:::wk;nyyI;G:;:j::>:B>N: F:nyyyyy]::yyyyyy::::::::::::::::::::::::::::::::::::::::::::::::::::: ::::::::::::::::::fyyyyya:nYf::G:I:K:wAyA::::::::::::::::::::::::::::: :::::::::::::::::::::::::::::::::::::::::::::::::JcvGYMt>^:fBWMtNHm=;: ::::::n:;`:Z@[::JrQUD:<::::::JDJ:j:VBYmp>HYLkNG>::::::::N:;::::::::_Z:vyyuy:>:<::::::AJ:n:>:nYN::wyyyq:[Z:F:i:k:m:o:q:s:u:w:y:;C:R<:TnEj``pkDqqHqqTP tWdZfbk;>ZEZ?GHZhV@cYH_WV>Z::j`>mKJVD cml>h:>H;B:AJ<>>eKP>>IllnGemnnGEm^JtJp>:[::AR=FZLVjrs:YZEF\\;NZBP@ZKj:DJ>===Zlb:KVGDJ=EKKDk;]AB=]R:?:;j?DJ=E;`A::;R=?b[ QcBdJ;@RJN\\=`F_rZoM;@B^FFI?RZJlJv>ZEFZ<>kjj:Hj@J;\\rVNZ@>Wlj:gml>Z:>::::::;C:?B:yay=J:>Z::::::J<>:QS:>Z:J;vCSml>d:NHEms> @[C:>Z::::::::kJ;@j;>:C:yayA:<::::::M:ke;B:FZ:vCJ::<::::::::=b:D=;B:]ZG::^rSjST:gc;=>:oi:;JBJwKJ p;:V<^vK>;NY^[:>Z:>:^:f??jd;Nh;f:Vd::V<>n>^=Vy<>rwM:<:sg:>Z<^ZaTXUe RYEU^dSN[RAglQ;N`D^Z>FI=Z:Nxs?;N@Fp>:jPf:CB:>x;^Z::_cJSNp;vq>:jP@J<^Z:Jrg:K:_c<;v:Sr==J:V_s_=q^lZKJ::Uk<^:>XCJ>JSNPQn:jS`Q>JSNECq:jS:_KSG_JSNE=h:=:[ SXK:_kK_b?N:;Zo>;bJS^tZ:VY;sy>Z:JB[:^ :^D:i=:FR>>:jP^:C:[q<>;N`DF`;V[>F:RGK:dJK@jJaj:Zo:d:_w;=:PMHjw?vx]:JBG :^::::jZKJ::Uk<^:>XCJ>JSdJSgi=F:>io>;N@AOA`j:jvPM>JSJLXj:Jv^=L J<::::=@;;:f?CJJSr:wc>?:CeXKJ:N@ndJFZ :jvdQ><:_;pj:JvdQHjw;<:s?JSJDIk:<:;;N @^vKF:>>OJ:N@nfDf:>^Z>>JSJnMj:JJN[:>:G;Ojysy=:;JHjw?:s?;B:=J;D lc`qsLqlp`_o\\DnF;B:OjJNk;Z:^Z:JZ:>::::WTJWTL>Z:Ja H>YtJ>d=C:]tJatJetJitJ>e=kfH>otJVe=KhH>utJytJ>f=kj H>?uJVf=KlH>EuJnf=kmH>KuJFg=KoH>QuJ^g=kpH>WuJvg=KrH>]uJNh=ksH>cuJfh=Ku H>iuJ>i=kvH>ouJVi=KxHN:fi=KyH>yuJ>Z:J:y=<><>;FJ>n:KZ=>;IJ>Z;KB@>;SJ>`; KZ@>[TJ>f;Kr@>;VJ>k;KBA>kVJ>n;KNAN:>p;KVA>;XJ>s;KbA>kXJ;JjA>KYJ>x;KvA6 >" }{TEXT -1 65 ". The following command attempts to compute this trip le integral." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "Tripleint(sq rt(x^2+y^2), z = 0..25 - x^2 - y^2, y = 0..sqrt(9 - x^2), x = -3..3); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$-F$6$*$,&*$)%\"xG\" \"#\"\"\"F0*$)%\"yGF/F0F0#F0F//%\"zG;\"\"!,(\"#DF0F,!\"\"F1F;/F3;F8*$, &\"\"*F0F,F;F4/F.;!\"$\"\"$" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$intG6$,.*&#\"\"$\"# ;\"\"\"*&)%\"xG\"\"%F+-%#lnG6#*$)F.\"\"#F+F+F+F+*&#\"#DF/F+*&F4F+F0F+F +!\"\"*(\"$B\"F+F/F:,&\"\"*F+F3F:#F+F5F+*&#F8F5F+*&F4F+-F16#,&*$F=F?F+ F)F+F+F+F+**F>F+\"\")F:F.F5F=F?F:*&#F)FHF+*&F-F+FCF+F+F:/F.;!\"$F)" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 192 "Maple does not have the capabili ty to evaluate this integral in rectangular form. However, it can be d one if the integral is converted to cylindrical coordiinates. In cylin drical coordinates, " }{XPPEDIT 18 0 "r = sqrt(x^2+y^2);" "6#/%\"rG-%% sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yGF+F," }{TEXT -1 40 ". The variable z still varies from 0 to " }{XPPEDIT 18 0 "25-x^2-y^2;" "6#,(\"#D\"\"\" *$%\"xG\"\"#!\"\"*$%\"yGF(F)" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "25-r^2 ;" "6#,&\"#D\"\"\"*$%\"rG\"\"#!\"\"" }{TEXT -1 7 ". For " }{TEXT 283 1 "x" }{TEXT -1 5 " and " }{TEXT 284 3 "y, " }{TEXT -1 31 "we can grap h the region in the " }{TEXT 285 3 "x-y" }{TEXT -1 58 " plane where th e region spans using the following command:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "plot(sqrt(9-x^2), x = -3..3, view = [-3..3, -3..3] );" }}{PARA 13 "" 1 "" {GLPLOT2D 476 476 476 {PLOTDATA 2 "6%-%'CURVESG 6$7_o7$$!\"$\"\"!$F*F*7$$!35+v=ng#=*H!#<$\"3y^\\;(*)pI@#!#=7$$!3A+]PM@ l$)HF/$\"3_Bx.gzhFJF27$$!3))*\\i:?ya(HF/$\"3A?v%G#o\"z#QF27$$!3)****\\ (oUInHF/$\"3*HP7xUtqT%F27$$!3u**\\7.k&4&HF/$\"3d@3O!psBS&F27$$!3&***** \\P&3Y$HF/$\"3AuRf\"R[&HiF27$$!3#****\\i!G\">!HF/$\"3o=:R%Rh&3wF27$$!3 !******\\2<#pGF/$\"3/gevX1Ch()F27$$!3')**\\78.K7GF/$\"3['G>6yRW/\"F/7$ $!3#)***\\7bBav#F/$\"3c>W7(\\Sk=\"F/7$$!3'***\\(=>P9p#F/$\"3h3unvS?D8F /7$$!36++]K3XFEF/$\"3+iBM*GIzW\"F/7$$!3%)****\\F)H')\\#F/$\"3q*o?x:w.m \"F/7$$!3#****\\i3@/P#F/$\"33u**o?yxQ=F/7$$!3;++Dr^b^AF/$\"3#3)[l!p[D) >F/7$$!3$****\\7Sw%G@F/$\"3)y'**yj+99@F/7$$!3*****\\7;)=,?F/$\"3ZF5H&o /]B#F/7$$!3/++DO\"3V(=F/$\"3$=P_g,GCM#F/7$$!3#******\\V'zV9TCF/7$$!3******\\d;%)G;F/$\"3c=_Lp/I>DF/7$$!3!******\\!)H%*\\\"F/$\" 3%Q>u,L0%)f#F/7$$!3/+++vl[p8F/$\"3]k:ICriUC\"F/$ \"3@')yC]1!)HFF/7$$!3-++DhkaI6F/$\"3a$3\\vLC)yFF/7$$!3s******\\XF`**F2 $\"3%*=yXzZ2IGF/7$$!3u*******>#z2))F2$\"3-tVvy9znGF/7$$!3S++]7RKvuF2$ \"33>LtDNP0HF/7$$!3s,+++P'eH'F2$\"3)Rd)eqH>LHF/7$$!3q)***\\7*3=+&F2$\" 3CQ)e'4$4!eHF/7$$!3[)***\\PFcpPF2$\"3!4r'[/JAwHF/7$$!3;)****\\7VQ[#F2$ \"3G/asc)*p*)HF/7$$!32)***\\i6:.8F2$\"3;P#\\`Kor*HF/7$$!3Wb+++v`hH!#?$ \"3Q$e:#Q&)****HF/7$$\"3]****\\(QIKH\"F2$\"3!=A'z&H6s*HF/7$$\"38****\\ 7:xWCF2$\"3_&GZ=\"4&*[eHF/7$$\"3A)****\\d6.B'F2$\"3AofqNBfMHF/7$$\"3 s****\\(o3lW(F2$\"3.j7G)R8h!HF/7$$\"35*****\\A))oz)F2$\"3q/UfVh7oGF/7$ $\"3e******Hk-,5F/$\"3v;uQ***yj<0\"GFF/7$$\"3M+++b*=jP\"F/$\"33x`nV0mlEF /7$$\"3g***\\(3/3(\\\"F/$\"3I2>I_'f(*f#F/7$$\"33++vB4JB;F/$\"3>b(RAunG _#F/7$$\"3u*****\\KCnu\"F/$\"3O3E;#zZ!RCF/7$$\"3s***\\(=n#f(=F/$\"3^cW fz>8TBF/7$$\"3P+++!)RO+?F/$\"3z#Hg0PUdB#F/7$$\"30++]_!>w7#F/$\"3.l6YwG +:@F/7$$\"3O++v)Q?QD#F/$\"31XU3XJ(*z>F/7$$\"3G+++5jypBF/$\"3Ig$)QPefR= F/7$$\"3<++]Ujp-DF/$\"31OSB9.Ca;F/7$$\"3++++gEd@EF/$\"3=&pI97Y&e9F/7$$ \"31+]PMh%\\o#F/$\"3/P6)y[1$Q8F/7$$\"39++v3'>$[FF/$\"3E64evZ!G?\"F/7$$ \"39+++5h(*3GF/$\"31%>:[y+M0\"F/7$$\"37++D6EjpGF/$\"3CT[z%o?wu)F27$$\" 3))*\\P%eWA-HF/$\"3u!*[p5]m'f(F27$$\"31+]i0j\"[$HF/$\"3YE5eEGv>iF27$$ \"3%*\\(=#HA6^HF/$\"377Li2S&QR&F27$$\"3E+D\"G:3u'HF/$\"3$>(4x`!*35WF27 $$\"3Uv$4Y6cb(HF/$\"3R_v`&ob=#QF27$$\"39]iSwSq$)HF/$\"3W2$[S()eE7$F27$ $\"3%[7.#Q?&=*HF/$\"3Lj$[z4c&4AF27$$\"\"$F*F+-%'COLOURG6&%$RGBG$\"#5! \"\"F+F+-%+AXESLABELSG6$Q\"x6\"Q!F\\al-%%VIEWG6$;F(F_`lFaal" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 6 "Thus, " }{TEXT 286 2 "r " }{TEXT -1 26 "wi ll vary from 0 to 3 and " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 10 " for 0 to " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 64 ". Thus , the triple integral can be represented (note that extra " }{TEXT 287 1 "r" }{TEXT -1 15 ") introduced as" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "Tripleint(r*r, z = 0..25 - r^2, r = 0..3, theta = 0.. Pi);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$-F$6$*$)%\"rG\" \"#\"\"\"/%\"zG;\"\"!,&\"#DF.F*!\"\"/F,;F2\"\"$/%&thetaG;F2%#PiG" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "This is the integral result" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(%);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#,$*(\"$#))\"\"\"\"\"&!\"\"%#PiGF&F&" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 "Exercises" }}{PARA 0 "" 0 "" {TEXT -1 89 "1. Sketch the three dimensi onal graph and find the contourplot of the following surfaces:" }} {PARA 0 "" 0 "" {TEXT -1 7 " a. " }{XPPEDIT 18 0 "f(x,y) = x*y*exp( -x^2/2-y^2/2);" "6#/-%\"fG6$%\"xG%\"yG*(F'\"\"\"F(F*-%$expG6#,&*&F'\" \"#F0!\"\"F1*&F(F0F0F1F1F*" }{TEXT -1 24 " for the interval -6 <= " } {TEXT 288 1 "x" }{TEXT -1 13 " <= 6, -6 <= " }{TEXT 289 1 "y" }{TEXT -1 5 " <= 6" }}{PARA 0 "" 0 "" {TEXT -1 7 " b. " }{XPPEDIT 18 0 "g( x,y) = sin(x)-sin(y);" "6#/-%\"gG6$%\"xG%\"yG,&-%$sinG6#F'\"\"\"-F+6#F (!\"\"" }{TEXT -1 20 " for the interval " }{XPPEDIT 18 0 "-4*Pi;" "6 #,$*&\"\"%\"\"\"%#PiGF&!\"\"" }{TEXT -1 4 " <= " }{TEXT 292 1 "x" } {TEXT -1 4 " <= " }{XPPEDIT 18 0 "4*pi;" "6#*&\"\"%\"\"\"%#piGF%" } {TEXT -1 2 ", " }{XPPEDIT 18 0 "-4*Pi" "6#,$*&\"\"%\"\"\"%#PiGF&!\"\" " }{TEXT -1 4 " <= " }{TEXT 291 1 "y" }{TEXT -1 4 " <= " }{XPPEDIT 18 0 "4*Pi;" "6#*&\"\"%\"\"\"%#PiGF%" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "2. Find the equation of t he tangent plane and normal line to the surface " }{XPPEDIT 18 0 "z^2+ y^2+z^2 = 9;" "6#/,(*$)%\"zG\"\"#\"\"\"F)*$)%\"yGF(F)F)*$F&F)F)\"\"*" }}{PARA 0 "" 0 "" {TEXT -1 96 " at the point (0, 0, 3). Plot the su rface, tangent plane, and normal line on the same graph. " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 45 "3. Evaluate the fo llowing multiple integrals." }}{PARA 0 "" 0 "" {TEXT -1 7 " a. 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