00020001010008main.ACT0002020012Area_TwoCurves.EAC010000002d80 Area_TwoCurves.EAC main.ACT#'  E 't[4#Riemann Sum and s between x+\3 AuthortN Wei-Chi Yang Radford University / e- l: wyang@r2.eduE!URL: http://www.#/5 Objectls:[T1. We use the dynamic geometry+CAS features within ClassPad (CP) to explorF L.[#@2. We explore left end, righ " Riemman Sums formulae with CP.[I[ Example 1.c  Find the area bounded by y=x2+2, -x, x=-1 and 2.\ 'Graphs of=^2.,/Ј(N\FinaForm$NGraph2D, 3@ LISTSYSL4< Modify P<STATCALC d< \x S:equence,xSheetO4| olveEq`wr0(Up<tupFLG1H(<Lis{pDPicViewWind_osvev'xx3ixҌh}y*|(y2<P \*  k1htȒԒ   ,Z!@("T4 #h@ $|L %X &d'p  H|u)*+,-.Ć0І1܆2345 E F,$ H@%%I<Jh2KTL` Ml.Nxdž OPQRSU]y ̆ ^_Ԇq؆ab䒌 ͇Α,)בT k,ّ|8 ڑD iۑ " FinancialFormat  ! system]list^]=_`a #b~uŽ @ @x8V aF seq_hbNewFolde]3 0 M<# , `   e5y  <Z [Note. R We describe the constructi  forfollowing anima (in Geometry Strip) as Y2s:w!step 1.rragnequMs y=x2/+2,-x,x=-1,{ d x=2 into~zbexwQw2wfiBinterse betweenjd-1 at Ar0g726B.[\Jstep 3. We animat point C along AB d choose the 5size to be n=30.[US4S construct7vertical line (say L) passes]rough{rparallelf\x=1iDi5i ersectionEtween}Ly=x2+2. [\PlY#V=>Έ$(6S29C `u`! tDEvKr  A 'LC uxy < E9  N2@U6\& I)h& @Pis D Coord of C: $ oCoA74Rc YoOG1#YpBE¶jHEcco 1@ l !  x^2+2 H )@ !7 ? 7line L@T6[O8aA74R~YL 1ŘbH"DYY`e؈]"#ȹFD!@ \3:'+5HE` #Z ge !H  *A@16&` BCCBC7 `C@ Ƙo y1D֘Du i$Rp^xH!^5"#܈F{HԈ7d     [8 Method 1.  We use Trapezoidal Sum& Approximation:[.WAfter the an we collectdistance EC as fows0d'py pasteGis into XSpreadsheet Editor.[2 1.90725327 1.835909631 785969084$ 7574316290297265H 764565993Z0023781}l5731272$9357 2.035671819 2.156956005 299643282 2.4637336# 649227111G856123665 3.084423306 3.33412604# 605231867589774078k 4.211652796 4.54696789$ 903686088 5.28180737Ć68133174$ 6.10225921H6Z45 7.00832342$ 7.493460168[LUWe note that sinc=30,e length of each subinterval is"dista5 between x=-2 -4and x=1 divides by 29, which is 0.1034482759.R 3/29R" )Du!. 44[\T 1The estimate}#listed under cell C1=10.50535078.AEcN SpreadsheetS   @ @ :Y#rS'Y cxYif6ut1bQu&PlvEe07 sr6Wr VqiV)C(. F73eDd'a#fD#0`C3A&`R1p6w@xPQ!Ry@lTig66(7 h1t2 "Y!TE#BI4`D `<Fit range->type B1:7->press enter.%Csum{q-' Method 2.6# set up Riemann Sum F̐qStep 1.2defines function.Rdefine f(x)=x^2+2+xRdoneR/[7 &Step 2. We Bthe right end point..Bqr(a,b,j,n)=a+j*(b-a)/n{%{3{lefz!zlz(j-1)~$~4sum6hr Q(f((n))1 rsum(-1,2,29)R" 10.8156956iVG.R4"[ "Step 5. We define the left end a66#lx a,b,n)=Q(f(lj )*((b-a)/(n))1$BdoneR K 19500595YE1[ Method 3.!set uparea analytically.Answer ! (x^2+2)-(-x)4-1=2Ext5]n ,Exercise 1. Έrv|) `@Gs$ . " DvKr  A 'L'C uxy qq:@  H B@\6r8W &1WG4z L UB &1WG4Y!(  1A@86Rcxt7 7^"G@ GYT"#9*0q6)& `bRFT Eq: $g ggf  %H B {`&B6QN '@  %$(&~)L9<0@\6c7n Stu4Y2~ Eq: *#NuV +1ǙxH,#-IȐc &1WG4YF.\!`\/X X -09H  !'+/1"5H!"$Pk }&1WG7 Z('$%&*. !1)#5qRsolve(x=3-(x-1)^2,x)R! x=-1,x=25[?!Therefore the intersctions (-1 2; and &2- respePvely.[Horizontal strip is easierR& (3-y^2)-(y+1)-21%yR39;2IH[$ Exercise 2. eFind the area bounded by x=yO,-y+6, a6x-axis.[Hint:\ GraphsЈ2N& %FinaForm$NGraph2D% 3 LISTSYS$@4< Modify XP<STATCALC |d< \x Sequence,xSheetO | olveEq`wr(UptupFLG1 (<Lis{HDPicViewWind_osvev ̐xy^(4؈(Uy2i(<|P $,  m0H<HT`lx ̆!؆"# ^W9%h &|r' !(,)8  *NDR+P,\-h.t012345EȆFԆHIJKT L M| N( O@ QLRRVXSdTp]` |i ^_`ab   ͆ΆІĆג ؒ܆نچ+ۑ| FinancialFormat  ) system]list^]=_`a #b~uŽ @ @x8V a seq_hb NewFolde53 0 pM< < `   e5y  <Z [Note. R We describe the constructi  forfollowing anima (in Geometry Strip) as Y2s:w!step 1.rragnequMs y=x2/+2,-x,x=-1,{ d x=2 into~zbexwQw2wfiBinterse betweenjd-1 at Ar0g726B.[\Jstep 3. We animat point C along AB d choose the 5size to be n=30.[US4S construct7vertical line (say L) passes]rough{rparallelf\x=1iDi5i ersectionEtween}Ly=x2+2. [\PlY#V=>Έ$(6S29C `u`! tDEvKr  A 'LC uxy < E9  N2@U6\& I)h& @Pis D Coord of C: $ oCoA74Rc YoOG1#YpBE¶jHEcco 1@ l !  x^2+2 H )@ !7 ? 7line L@T6[O8aA74R~YL 1ŘbH"DYY`e؈]"ȹFD!@ \3:'+5HE` #Z ge !H  *A@16&` BCCBC7 `C@ Ƙo y1D֘Du i$Rp^xH!^5"#܈F{HԈ7d     [8 Method 1.  We use Trapezoidal Sum& Approximation:[.WAfter the an we collectdistance EC as fows0d'py pasteGis into XSpreadsheet Editor.[2 1.90725327 1.835909631 785969084$ 7574316290297265H 764565993Z0023781}l5731272$9357 2.035671819 2.156956005 299643282 2.4637336# 649227111G856123665 3.084423306 3.33412604# 605231867589774078k 4.211652796 4.54696789$ 903686088 5.28180737Ć68133174$ 6.10225921H6Z45 7.00832342$ 7.493460168[LUWe note that sinc=30,e length of each subinterval is"dista5 between x=-2 -4and x=1 divides by 29, which is 0.1034482759.R 3/29R" )Du!. 44[\T 1The estimate}#listed under cell C1=10.50535078.AEcN SpreadsheetS   @ @ :Y#rS'Y cxYif6ut1bQu&PlvEe07 sr6Wr VqiV)C(. F73eDd'a#fD#0`C3A&`R1p6w@xPQ!Ry@lTig66(7 h1t2 "Y!TE#BI4`D `<Fit range->type B1:7->press enter.%Csum{q-' Method 2.6# set up Riemann Sum F̐qStep 1.2defines function.Rdefine f(x)=x^2+2+xRdoneR/[7 &Step 2. We Bthe right end point..Bqr(a,b,j,n)=a+j*(b-a)/n{%{3{lefz!zlz(j-1)~$~4sum6hr Q(f((n))1 rsum(-1,2,29)R" 10.8156956iVG.R4"[ "Step 5. We define the left end a66#lx a,b,n)=Q(f(lj )*((b-a)/(n))1$BdoneR K 19500595YE1[ Method 3.!set uparea analytically.Answer ! (x^2+2)-(-x)4-1=2Ext5]n ,Exercise 1. Έrv|) `@Gs$ . " DvKr  A 'L'C uxy qq:@  H B@\6r8W &1WG4z L UB &1WG4Y!(  1A@86Rcxt7 7^"G@ GYT"#9*0q6)& `bRFT Eq: $g ggf  %H B {`&B6QN '@  %$(&~)L9<0@\6c7n Stu4Y2~ Eq: *#NuV +1ǙxH,#-IȐc &1WG4YF.\!`\/X X -09H  !'+/1"5H!"$Pk }&1WG7 Z('$%&*. !1)#5qRsolve(x=3-(x-1)^2,x)R! x=-1,x=25[?!Therefore the intersctions (-1 2; and &2- respePvely.[Horizontal strip is easierR& (3-y^2)-(y+1)-21%yR39;2IH[$ Exercise 2. eFind the area bounded by x=yO,-y+6, a6x-axis.[Hint:\ GraphsЈ2N& %FinaForm$NGraph2D% 3 LISTSYS$@4< Modify XP<STATCALC |d< \x Sequence,xSheetO | olveEq`wr(UptupFLG1 (<Lis{HDPicViewWind_osvev ̐xy^(4؈(Uy2i(<|P $,  m0H<HT`lx ̆!؆"# ^W9%h &|r' !(,)8  *NDR+P,\-h.t012345EȆFԆHIJKT L M| N( O@ QLRRVXSdTp]` |i ^_`ab   ͆ΆІĆג ؒ܆نچ+ۑ| FinancialFormat  ) system]list^]=_`a #b~uŽ @ @x8V a seq_hb NewFolde53 0 pM< <ǧrc/!KQ\JHM'vR\qph,'8jMoz=O]jޫ"'ԁCzj8#]߷kCb]ߺmqM嘵7[XnK"2w9\TX,Zn(ZN-NjcEѢŊ+j_}Ŋ+j_}Ŋ+j_}ѢE-j_}ѢE-j_}Ѣ5MEˍEɢOrh9VU9,2,yv .krn M7нi~*۪i6E#Te%~dl.c~nB>}=}[+wI`N~lŠYއմd[cY{taR}n|Hd]\l#Ewz.cP*5:8i˭Tzw@<eB6;26۷;zoGĎX=ýW>Ɓ=}f^k?j_˭T7T,boXPԾeCXH[vTlU*VbU*Qe*֢bUMŪUSjTKŮQ~[b*v=b*QT츊]bϨJ;bu*ݠb؍*v^nRի[b͹ؒ2[b! 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