0002000201000fRiemann_Sum.ACT000102000fRiemann_Sum.EAC0100000024cf  Final.EAC main.ACT&)-@ `I 'K[4Applications of Riemann Sum#\+ AuthorBc Wei-Chi Yang Radford University / e- l: wyang@r2.eduE!URL: http://www.#/5 Objectts:[M1) We will use examples to demtrate that the ideausing2Z5approximate the area is not limited to when partition(s.allel'eiDr x or y axis.[T2) In o(words, lying Riemann Sumupcoordin free and \th$possible because ofintegraÈDynamic Geometry SystemSCompuťAlgebra#.[׈ Example 1.6 FiSboundL by y=x^2+3,-1,x=x=2X Soluj: j\P:when we divide into 19 panels,get 12.56509695=>see D19  'pFf `!gxR4`7DWvKr  A 'L'C uxy qq 1@! H!E@6&1WH 3$ Ux^2+3 >>H vTyo)&1WHY !@# ' 2"@A6@LW e@ e !wCf shBUUfj K5Ġ] !gf   l:H A` &CBL7  z @  H,Du- HD+&1WG7_  R^^!@6\\t`H! ܈d  !玳- '[LPstep 1. find the intersection points of two verical lines with y=x-1 at A aAB W respDvely.[l9$step 2. draw the line segment AB andopy=x-1.[9A 3. pick a poi8C7 imate alongYAFA4. construct apassesroughTd parallel to x=-1 or2.[OZO5.leLӌEat is ininters%ion ofvertical(from #4)  ^2+37u6. Wedžy distance Efterxu.[GKey:QQ? We are looking at a trapezoid in each panel. How do we find th=a of'seI?[WIf? sub-divide ABf to 19 (n=20)ms,_distance between tworallel lines (byw=usňmeasure) is 0.1578947368 oru'lengtABR 3/19R" CWG6! [:T followFcolumn{#lection)EC.v#4 3.7091412746814404$ 3.27700831 3.135734072 3.04432133#0065011080335G 069252078Y77285319k 335180055}542936286 40174.j4 4.46537396872576r 5.32963988~ 5.83656509$6.39 7[\0GApproximated area using trapezoidal rule is lis& under C1=12.o695W[wN SpreadsheetS' @$ @ =c1pA'2FD'p1g6W4 QC!36R03QRr13QPT)6(Tp)$?DFSs`%vQ29ee p9W1 `DD  !!J WG6 3-% aKK]KK@ Qm cTUU !gDKQ K@   ,8 wDD  !  WG6 !@i qqqwrrrrNl,@[PActual answer:R((x^2+3)-(x-1))1-1:2BxR-O13.5[[ Example 2 v<Find the area bounded by x^2+y2.22x-0.1y529=0 a70y=0.980& +1.866. SeeUGeometry Strip below.[8 Solution:[J UWe have twosspecificurves; one big small. F demonstrate *how to f-erinis case@noAdistance of EF=2.444009ubdivideGb 19 panelsR$ 1 /19R"; U286320526(c RcS #+.<1. Each subinterval is of length 0.1286320526.[$2. The#-HG6tabled below:\,play animationΈBFLQU] 1 `@Gs( ./ DvKr   A 'L'C uxy  @  G@,6s l`x@Q pJ  UBFd !% 3e!Yi H  H@%6,69ha& `3Y#CS\@ GYTڗj"2@!*Ii@hRg{ Y 42HYU)` 2@"!7 rA"C"z Be7YP[Ye#%H ! $@ "%4&JȐ0@]6dFbo Y!% 3e!Yi.`F'Řvv(v)!* # &6HY+TH E4f I$E4V#Uߐ~,CF(&v7'gb(v2B-. @! ,*&W2gf//@ /&05*Tp9j ]s &1WG7 V1H2d@6FYh!%td2gx YB3Ř+4\a\s2JB.-*5_2Ĉh6Ǡވ   8 #$."! " +,*42&@=0 [\0 0.2338395223 0.392711369 511848075760579856J 6753060238] 7294542441p768531&94005179_ 80653845999_Ѥ?0[dKV3. We use Trapezoidal Rule here by draggin the above values for HG into following \Spreadsheet Strip.\/Approximated area is lisunder C1=1.41912895Q8N SpreadsheetS  @ @ 5C"CBdb7H  s`(Q`?@  =<@/H!>A <آ abC#P M &1WG7t A}2hBH zFp6QCDQaY3Vy#.CCGC5V!`X$`DUE+Ȁ CYp D$wdFXXGX"gUYtAS !EU`C H H! BEI'H G"JE2KWH"JD@g6n(AyY f8qf U; Dline 376WtASDb gUn g0sCLV@ ^W^t^2^R"^jA Y#q'F LX!H"bY^1^B^Tq9PUW2gfZPƘ=+/[\ ]"`NbVbhYH ` W]^. +M@>6H@&2`CRZNEN  X`hY=U><;GKLBCI:[ Solutiion:Step 1. Find the intersect&s:= Psolve({x^2+y4.25x+1.2y+0.4727=0, 0.9 x-0.94y3604=0},{x,y}) Rx=-" 0.04569462689 +,y, 0.3181948096H Ur , x=-" 1.004840268H@&ya44,y=) 1.17430752CQ\Rb`[i*SStep 2. Pick one of the intersection M=[-},`] and construct; 1tangent liVforWsmall circle a0pom M: )solve(x^2+y0.92x-4y3604=0,y)R-y=C-  -10000Ca2 -92+5813-477^Q2 -9200x+5813+47100R/. tanLine( -%CT\,x,-1.004840268)RdP 287601234661962144708854534297Ќ;218882173698424810+`b3[ 25077572881.)+ 457646847506f [ OStep 3. Drag the o"r intersections N=[-0.04569462689, 3181948096]/o>Vgeometry strip.nXn4. CVtruct a parallel linessin rough N and$gtangent3atwMjYj5ja erpendicularP s but notsm circle:[G-picko: (A) on1#B-chD1 A.m6=Stpe 6. Find the intersection (B) between line 3 a( 2. [D6Step 7. ConstructJ1 segment AB:hid3.[?V?8. Pick a point (C) 8animate C alongO/steps of&to be 20.Mk9. Tlength8 AB=1.770858each sub val divinX 19 panels:RA @ /19R" 0.09320305263 2&1Xw <?G distance DG[!0 0.4199356922{ 0.5979567908 7242162496 0.81953369$ 89194931194553278& 9826047541] 1.00454220K1212791$571264I52823203 95046190759]8186o83391䐂 748611061_64076J 503207180_ 0.31884738K0\34KThe approximated area using Trapezoidal rule is lis&under C1=1.331466824剆_N SpreadsheetSI @(B @ <"6V  g$!bI`6S6gQl ES'`GTEB m !'Wd6 (# 0 PFP) E3YtHaa` ꊴ @vQ qP sPh`<see D19  'pFf `!gxR4`7DWvKr  A 'L'C uxy qq 1@! H!E@67ZY"V% Ux^2+3 H>H vTyomu   !@# ' 2"@A6HLW e@ e !wCfYGP2`Ufj K5Ġ] !gf`  :H A ` &CBL7zD @  H,Du- HD+&1WG7_  R^^!@6\\t`H! ܈d  !玳- '[LPstep 1. find the intersection points of two verical lines with y=x-1 at A aAB W respDvely.[l9$step 2. draw the line segment AB andopy=x-1.[9A 3. pick a poi8C7 imate alongYAFA4. construct apassesroughTd parallel to x=-1 or2.[OZO5.leLӌEat is ininters%ion ofvertical(from #4)  ^2+37u6. Wedžy distance Efterxu.[GKey:QQ? We are looking at a trapezoid in each panel. How do we find th=a of'seI?[WIf? sub-divide ABf to 19 (n=20)ms,_distance between tworallel lines (byw=usňmeasure) is 0.1578947368 oru'lengtABR 3/19R" CWG6! [:T followFcolumn{#lection)EC.v#4 3.7091412746814404$ 3.27700831 3.135734072 3.04432133#0065011080335G 069252078Y77285319k 335180055}542936286 40174.j4 4.46537396872576r 5.32963988~ 5.83656509$6.39 7[\0GApproximated area using trapezoidal rule is lis& under C1=12.o695W[wN SpreadsheetS' @$ @ =c1pA'2FD'p1g6W4 QC!36R03QRr13QPT)6(Tp)$?DFSs`%vQ29ee p9W1 `DD  !!J WG6 3-% aKK]KK@ Qm cTUU !gDKQ K@   ,8 wDD  !  WG6 !@i qqqwrrrrNl,@[PActual answer:R((x^2+3)-(x-1))1-1:2BxR-O13.5[[ Example 2 v<Find the area bounded by x^2+y2.22x-0.1y529=0 a70y=0.980& +1.866. SeeUGeometry Strip below.[8 Solution:[J UWe have twosspecificurves; one big small. F demonstrate *how to f-erinis case@noAdistance of EF=2.444009ubdivideGb 19 panelsR$ 1 /19R"; U286320526(c RcS #+.<1. Each subinterval is of length 0.1286320526.[$2. The#-HG6tabled below:\,play animationΈBFLQU] 1 `@Gs( ./ DvKr   A 'L'C uxy  @  G@,6s l`x@Q pJ  UBFd !% 3e!Yi H  H@%6,69ha& `3Y#CS\@ GYTڗj"2@!*Ii@hRg{ Y 42HYU)` 2@"!7 rA"C"z Be7YP[Ye#%H ! $@ "%4&JȐ0@]6dFbo Y!% 3e!Yi.`F'Řvv(v)!* # &6HY+TH E4f I$E4V#Uߐ~,CF(&v7'gb(v2B-. @! ,*&W2gf//@ /&05*Tp9j ]s &1WG7 V1H2d@6FYh!%td2gx YB3Ř+4\a\s2JB.-*5_2Ĉh6Ǡވ   8 #$."! " +,*42&@=0 [\0 0.2338395223 0.392711369 511848075760579856J 6753060238] 7294542441p768531&94005179_ 80653845999_Ѥ?0[dKV3. We use Trapezoidal Rule here by draggin the above values for HG into following \Spreadsheet Strip.\/Approximated area is lisunder C1=1.41912895Q8N SpreadsheetS  @ @ 5C"CBdb7H  s`(Q`?@  =<@/H!>A <آ abC#P M &1WG7t AhBH zGp6HfAWG)YmCȀ BYp D$wDEX"gUtASC F5FF\H\G"mDD'bq YyAI`Hʀ"@6&P%`7 Y $IR@  TGEJp!H"< EW2gftKE;EI &??LH".>MȎline 3tASD gU g0sNш^O^2/R"^jA/#q'F;LXQ'^1gU" tASB YTq9PMW2gfR2@  =MSPŠQ&T"U"`pN@6VbhYH $Y<;EGKZA:)[p Solutiion:Step 1. Find the intersect&s:u=P solve({x^2+y4.25x+1.2y+0.4727=0, 0.9 x-0.94y$3604=0},{x,y}) Rx=-"  0.04569462689Vbh# +,y, 0.3181948096H UrN +,a 1.004840`H@&ya,y=) 1.17430752CQ\R[*SStep 2. Pick one of the intersection M=[-},`] and construct; 1tangent liVforWsmall circle a0pom M:) solve(x^2+y0.92x-4y3604=0,y)Ry= -  -10000C)2 -92+5813-47O7Z,[Q+QR tanLine(\,x,-1.004840268%dP 287601S834661962144708854534297};218882173698424810+`}#3346619621447088545342977 250775728811+4100+ 457646847506fR[lOStep 3. Drag the o"r intersections N=[-0.04569462689, 3181948096]/o>geometry strip.nXn4. CVtruct a parallel linessin rough N and$gtangent3atwMjY,Step 5. Construct a line perpendicular to tharallel#s but not interse=%sm$ circle:[-pickc point (A) onp1#B-cs1 and passing through A.m =Stpe 6. Findion (B) betwee3O2. E6!7! # segment AB:hid3.[?V?8. PC8 animate C aloAB/rss of&]Lbe 20.[MStep 9. 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