{VERSION 6 0 "IBM INTEL NT" "6.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 1 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 1 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 1 } {CSTYLE "Blue Emphasis" -1 256 "Times" 1 12 0 0 255 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Green Emphasis" -1 257 "Times" 1 12 0 128 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Red Emphasis" -1 258 "Times" 1 12 255 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Maroon Emphasis" -1 259 "Times" 1 12 128 0 128 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Dark Red Emphasis" -1 260 "Times" 1 12 128 0 0 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Purple Emphasis" -1 261 " Times" 1 12 102 0 230 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "Grey Emphasis" -1 262 "Times" 1 12 96 52 84 1 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" 260 264 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 260 265 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 260 266 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 260 267 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 260 268 "" 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 1" -1 3 1 {CSTYLE "" -1 -1 "Times " 1 18 0 0 128 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 4 3 0 3 0 2 2 0 1 } {PSTYLE "Heading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 128 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Maple Output " -1 11 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 3 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Normal" -1 257 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }3 1 0 0 0 0 1 0 1 0 2 2 0 1 }{PSTYLE "Bullet Item " -1 258 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 } 3 1 0 0 3 3 1 0 1 0 2 2 15 2 }{PSTYLE "Normal" -1 259 1 {CSTYLE "" -1 -1 "Times" 1 12 0 0 0 1 2 2 2 2 2 2 1 1 1 1 }1 1 0 0 0 0 1 0 1 0 2 2 0 1 }} {SECT 0 {PARA 3 "" 0 "" {TEXT -1 31 "Integration using Riemann sums " }{TEXT 263 0 "" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 37 "by Peter Stone, Nanaimo, B.C., Canada" }}{PARA 0 "" 0 "" {TEXT -1 19 "Version: 22.3.2007 " }{TEXT 260 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "Summation formulas" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{PARA 0 " " 0 "" {TEXT -1 98 "The following summation formulas are required to e stablish the results given in the next section. " }}{SECT 1 {PARA 4 " " 0 "" {XPPEDIT 18 0 "Sum(i,i = 1 .. n) = n*(n+1)/2;" "6#/-%$SumG6$%\" iG/F';\"\"\"%\"nG*(F+F*,&F+F*F*F*F*\"\"#!\"\"" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "Sum(i,i=1..n)=factor(sum(i,i=1..n));\nS := unapply(rhs(%),n): 'S(n )'=S(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$%\"iG/F';\"\"\"% \"nG,$*(\"\"#!\"\"F+F*,&F+F*F*F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 " 6#/-%\"SG6#%\"nG,$*(\"\"#!\"\"F'\"\"\",&F'F,F,F,F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 181 "The following calcula tion gives numerical examples of this summation formula and also provi des a Maple proof of the inductive step needed to demonstrate the vali dity of the formula." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "mat rix([['n','Sum(i,i = 1 .. n)'],[`________`,`__________________________ _____________`],\nseq([n,S(n)],n=1..5),[` . `,` . `],[n,'S(n)'=S(n)],[ n+1,'S(n+1)'=S(n+1)],[` . `,` . `]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#7-7$%\"nG-%$SumG6$%\"iG/F,;\"\"\"F(7$%)________G%H_____ __________________________________G7$F/F/7$\"\"#\"\"$7$F6\"\"'7$\"\"% \"#57$\"\"&\"#:7$%$~.~GF@7$F(/-%\"SG6#F(,$*(F5!\"\"F(F/,&F(F/F/F/F/F/7 $FI/-FD6#FI,$*(F5FHFIF/,&F(F/F5F/F/F/F?Q)pprint246\"" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 94 "'S(n+1)-S (n)'=S(n+1)-S(n);\n``=``(expand(2*S(n+1)))/2-``(expand(2*S(n)))/2;\n`` =expand(rhs(%%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%\"SG6#,&%\"n G\"\"\"F*F*F*-F&6#F)!\"\",&*(\"\"#F-F(F*,&F)F*F0F*F*F**(F0F-F)F*F(F*F- " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"#F(-F$6#,(*$)% \"nGF)F(F(*&\"\"$F(F/F(F(F)F(F(F(*&#F(F)F(-F$6#,&F-F(F/F(F(!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&%\"nG\"\"\"F'F'" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "Sum(i^2,i = 1 .. n) = n*(n+1)*( 2*n+1)/6;" "6#/-%$SumG6$*$)%\"iG\"\"#\"\"\"/F);F+%\"nG**F.F+,&F.F+F+F+ F+,&*&F*F+F.F+F+F+F+F+\"\"'!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Sum(i^2,i=1. .n)=factor(sum(i^2,i=1..n));\nS := unapply(rhs(%),n): 'S(n)'=S(n);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*$)%\"iG\"\"#\"\"\"/F);F+%\" nG,$**\"\"'!\"\"F.F+,&F.F+F+F+F+,&*&F*F+F.F+F+F+F+F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"SG6#%\"nG,$**\"\"'!\"\"F'\"\"\",&F'F,F,F,F,,& *&\"\"#F,F'F,F,F,F,F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 181 "The following calculation gives numerical example s of this summation formula and also provides a Maple proof of the ind uctive step needed to demonstrate the validity of the formula." }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 227 "matrix([['n','n^2','Sum(i^2 ,i = 1 .. n)'],[`________`,`_________`,`______________________________ _________`],\nseq([n,n^2,S(n)],n=1..8),[` . `,` . `,` . `],[n,n^2,'S(n )'=S(n)],[n+1,(n+1)^2,'S(n+1)'=S(n+1)],[` . `,` . `,` . `]]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#707%%\"nG*$)F(\"\"#\"\"\"- %$SumG6$*$)%\"iGF+F,/F2;F,F(7%%)________G%*_________G%H_______________ ________________________G7%F,F,F,7%F+\"\"%\"\"&7%\"\"$\"\"*\"#97%F;\"# ;\"#I7%F<\"#D\"#b7%\"\"'\"#O\"#\"*7%\"\"(\"#\\\"$S\"7%\"\")\"#k\"$/#7% %$~.~GFTFT7%F(F)/-%\"SG6#F(,$**FH!\"\"F(F,,&F(F,F,F,F,,&*&F+F,F(F,F,F, F,F,F,7%Fgn*$)FgnF+F,/-FX6#Fgn,$**FHFfnFgnF,,&F(F,F+F,F,,&*&F+F,F(F,F, F>F,F,F,FSQ)pprint256\"" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "'S(n+1)-S(n)'=S(n+1)-S(n);\n``=``( expand(6*S(n+1)))/6-``(expand(6*S(n)))/6;\n``=expand(rhs(%%));\n``=fac tor(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%\"SG6#,&%\"nG\"\" \"F*F*F*-F&6#F)!\"\",&**\"\"'F-F(F*,&F)F*\"\"#F*F*,&*&F2F*F)F*F*\"\"$F *F*F***F0F-F)F*F(F*,&*&F2F*F)F*F*F*F*F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"'F(-F$6#,**&\"\"#F()%\"nG\"\"$F(F(*& \"\"*F()F0F.F(F(*&\"#8F(F0F(F(F)F(F(F(*&#F(F)F(-F$6#,(*&F.F(F/F(F(*&F1 F(F4F(F(F0F(F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,(*$)%\"nG \"\"#\"\"\"F**&F)F*F(F*F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G* $),&%\"nG\"\"\"F)F)\"\"#F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "Sum(i^3,i = 1 .. n) = n^2*( n+1)^2/4;" "6#/-%$SumG6$*$)%\"iG\"\"$\"\"\"/F);F+%\"nG*(F.\"\"#,&F.F+F +F+F0\"\"%!\"\"" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Sum(i^3,i=1..n)=factor(sum(i ^3,i=1..n));\nS := unapply(rhs(%),n): 'S(n)'=S(n);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#/-%$SumG6$*$)%\"iG\"\"$\"\"\"/F);F+%\"nG,$*(\"\"%!\" \"F.\"\"#,&F.F+F+F+F3F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"SG6#% \"nG,$*(\"\"%!\"\"F'\"\"#,&F'\"\"\"F.F.F,F." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 181 "The following calculation give s numerical examples of this summation formula and also provides a Map le proof of the inductive step needed to demonstrate the validity of t he formula." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 227 "matrix([['n' ,'n^3','Sum(i^3,i = 1 .. n)'],[`________`,`_________`,`_______________ ________________________`],\nseq([n,n^3,S(n)],n=1..8),[` . `,` . `,` . `],[n,n^3,'S(n)'=S(n)],[n+1,(n+1)^3,'S(n+1)'=S(n+1)],[` . `,` . `,` . `]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#707%%\"nG*$)F(\" \"$\"\"\"-%$SumG6$*$)%\"iGF+F,/F2;F,F(7%%)________G%*_________G%H_____ __________________________________G7%F,F,F,7%\"\"#\"\")\"\"*7%F+\"#F\" #O7%\"\"%\"#k\"$+\"7%\"\"&\"$D\"\"$D#7%\"\"'\"$;#\"$T%7%\"\"(\"$V$\"$% y7%F<\"$7&\"%'H\"7%%$~.~GFUFU7%F(F)/-%\"SG6#F(,$*(FB!\"\"F(F;,&F(F,F,F ,F;F,7%Fhn*$)FhnF+F,/-FY6#Fhn,$*(FBFgnFhnF;,&F(F,F;F,F;F,FTQ)pprint266 \"" }}}{PARA 4 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "'S(n+1)-S(n)'=S(n+1)-S(n);\n``=``(expand(4*S(n+1)))/ 4-``(expand(4*S(n)))/4;\n``=expand(rhs(%%));\n``=factor(rhs(%));" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%\"SG6#,&%\"nG\"\"\"F*F*F*-F&6#F)! \"\",&*(\"\"%F-F(\"\"#,&F)F*F1F*F1F**(F0F-F)F1F(F1F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"\"%F(-F$6#,,*$)%\"nGF)F(F(*&\"\"'F( )F/\"\"$F(F(*&\"#8F()F/\"\"#F(F(*&\"#7F(F/F(F(F)F(F(F(*&#F(F)F(-F$6#,( F-F(*&F7F(F2F(F(*$F6F(F(F(!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%! G,**$)%\"nG\"\"$\"\"\"F**&F)F*)F(\"\"#F*F**&F)F*F(F*F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*$),&%\"nG\"\"\"F)F)\"\"$F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}}{SECT 1 {PARA 4 "" 0 "" {XPPEDIT 18 0 "Sum(i^4,i = 1 .. n)=n*(n+1)*(2*n+1)*(3*n^2+3*n-1)/30" "6#/-%$Sum G6$*$%\"iG\"\"%/F(;\"\"\"%\"nG*,F-F,,&F-F,F,F,F,,&*&\"\"#F,F-F,F,F,F,F ,,(*&\"\"$F,*$F-F2F,F,*&F5F,F-F,F,F,!\"\"F,\"#IF8" }{TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Sum(i^4,i=1..n)=factor(sum(i^4,i=1..n));\nS := unapply(rhs(%),n): \+ 'S(n)'=S(n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$SumG6$*$)%\"iG\"\" %\"\"\"/F);F+%\"nG,$*,\"#I!\"\"F.F+,&F.F+F+F+F+,&*&\"\"#F+F.F+F+F+F+F+ ,(*&\"\"$F+)F.F6F+F+*&F9F+F.F+F+F+F2F+F+" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%\"SG6#%\"nG,$*,\"#I!\"\"F'\"\"\",&F'F,F,F,F,,&*&\"\"#F,F'F,F, F,F,F,,(*&\"\"$F,)F'F0F,F,*&F3F,F'F,F,F,F+F,F," }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 181 "The following calculatio n gives numerical examples of this summation formula and also provides a Maple proof of the inductive step needed to demonstrate the validit y of the formula." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 227 "matrix ([['n','n^4','Sum(i^4,i = 1 .. n)'],[`________`,`_________`,`_________ ______________________________`],\nseq([n,n^4,S(n)],n=1..8),[` . `,` . `,` . `],[n,n^4,'S(n)'=S(n)],[n+1,(n+1)^4,'S(n+1)'=S(n+1)],[` . `,` . `,` . `]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#K%'matrixG6#707%%\"nG* $)F(\"\"%\"\"\"-%$SumG6$*$)%\"iGF+F,/F2;F,F(7%%)________G%*_________G% H_______________________________________G7%F,F,F,7%\"\"#\"#;\"#<7%\"\" $\"#\")\"#)*7%F+\"$c#\"$a$7%\"\"&\"$D'\"$z*7%\"\"'\"%'H\"\"%vA7%\"\"( \"%,C\"%wY7%\"\")\"%'4%\"%s()7%%$~.~GFVFV7%F(F)/-%\"SG6#F(,$*,\"#I!\" \"F(F,,&F(F,F,F,F,,&*&F;F,F(F,F,F,F,F,,(*&F?F,)F(F;F,F,*&F?F,F(F,F,F,F inF,F,7%Fjn*$)FjnF+F,/-FZ6#Fjn,$*,FhnFinFjnF,,&F(F,F;F,F,,&*&F;F,F(F,F ,F?F,F,,(*&F?F,)FjnF;F,F,*&F?F,F(F,F,F;F,F,F,FUQ)pprint276\"" }}} {PARA 4 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 117 "'S(n+1)-S(n)'=S(n+1)-S(n);\n``=``(expand(30*S(n+1)))/30-``(expand (30*S(n)))/30;\n``=expand(rhs(%%));\n``=factor(rhs(%));" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#/,&-%\"SG6#,&%\"nG\"\"\"F*F*F*-F&6#F)!\"\",&*,\" #IF-F(F*,&F)F*\"\"#F*F*,&*&F2F*F)F*F*\"\"$F*F*,(*&F5F*)F(F2F*F**&F5F*F )F*F*F2F*F*F**,F0F-F)F*F(F*,&*&F2F*F)F*F*F*F*F*,(*&F5F*)F)F2F*F**&F5F* F)F*F*F*F-F*F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,&*&#\"\"\"\"#IF (-F$6#,.*&\"\"'F()%\"nG\"\"&F(F(*&\"#XF()F0\"\"%F(F(*&\"$I\"F()F0\"\"$ F(F(*&\"$!=F()F0\"\"#F(F(*&\"$>\"F(F0F(F(F)F(F(F(*&#F(F)F(-F$6#,**&F.F (F/F(F(*&\"#:F(F4F(F(*&\"#5F(F8F(F(F0!\"\"F(FJ" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G,,*$)%\"nG\"\"%\"\"\"F**&F)F*)F(\"\"$F*F**&\"\"'F*) F(\"\"#F*F**&F)F*F(F*F*F*F*" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G*$) ,&%\"nG\"\"\"F)F)\"\"%F)" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}} }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 31 "Integration using Riem ann sums " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "Int(f(x),x=a..b)" "6#-%$IntG6$-%\"fG6#%\"xG/F); %\"aG%\"bG" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 20 " Subdividin g points " }{XPPEDIT 18 0 "x[0]=a,x[1],x[2],` . . . `,x[n]=b" "6'/&%\" xG6#\"\"!%\"aG&F%6#\"\"\"&F%6#\"\"#%(~.~.~.~G/&F%6#%\"nG%\"bG" }{TEXT -1 8 ", where " }{XPPEDIT 18 0 "x[i]-x[i-1] = h;" "6#/,&&%\"xG6#%\"iG \"\"\"&F&6#,&F(F)F)!\"\"F-%\"hG" }{XPPEDIT 18 0 "``=(b-a)/n" "6#/%!G*& ,&%\"bG\"\"\"%\"aG!\"\"F(%\"nGF*" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x[0]=a, x[1]=a+h,x[2]=a+2*h,` . . . `, x[i]=a+i*h,` . . . `,x[n]=a+n*h" "6)/&%\"xG6#\"\"!%\"aG/&F%6#\"\"\",&F (F,%\"hGF,/&F%6#\"\"#,&F(F,*&F2F,F.F,F,%(~.~.~.~G/&F%6#%\"iG,&F(F,*&F9 F,F.F,F,F5/&F%6#%\"nG,&F(F,*&F?F,F.F,F," }{XPPEDIT 18 0 "``=b" "6#/%!G %\"bG" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{XPPEDIT 18 0 "x[i]=a+i*h" "6#/&%\"xG6#%\"iG,&%\"aG\"\"\"*&F'F*%\"hGF*F*" } {XPPEDIT 18 0 "`` = a+i*``((b-a)/n);" "6#/%!G,&%\"aG\"\"\"*&%\"iGF'-F$ 6#*&,&%\"bGF'F&!\"\"F'%\"nGF/F'F'" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 2 " " }{TEXT 257 8 "Left sum" }{TEXT -1 5 ": " }{XPPEDIT 18 0 "Sum(f(x[i])*h,i=0..n-1)=Sum(f(a+i*``((b-a)/n))*``((b-a)/n),i=0.. n-1)" "6#/-%$SumG6$*&-%\"fG6#&%\"xG6#%\"iG\"\"\"%\"hGF//F.;\"\"!,&%\"n GF/F/!\"\"-F%6$*&-F)6#,&%\"aGF/*&F.F/-%!G6#*&,&%\"bGF/F=F6F/F5F6F/F/F/ -F@6#*&,&FDF/F=F6F/F5F6F//F.;F3,&F5F/F/F6" }{TEXT -1 1 "." }}{PARA 0 " " 0 "" {TEXT -1 2 " " }{TEXT 257 9 "Right sum" }{TEXT -1 5 ": " } {XPPEDIT 18 0 "Sum(f(x[i])*h,i=1..n)= Sum(f(a+i*``((b-a)/n))*``((b-a)/ n),i = 1 .. n)" "6#/-%$SumG6$*&-%\"fG6#&%\"xG6#%\"iG\"\"\"%\"hGF//F.;F /%\"nG-F%6$*&-F)6#,&%\"aGF/*&F.F/-%!G6#*&,&%\"bGF/F:!\"\"F/F3FBF/F/F/- F=6#*&,&FAF/F:FBF/F3FBF//F.;F/F3" }{TEXT -1 2 ". " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{TEXT 257 10 "Middle sum" }{TEXT -1 2 ": " }{XPPEDIT 18 0 "Sum(f((x[i]+x[i-1])/2)*h,i = 1 .. n) = Sum(f(a+(2*i-1)*``((b-a)/ (2*n))*``((b-a)/n),i = 1 .. n)" "6#/-%$SumG6$*&-%\"fG6#*&,&&%\"xG6#%\" iG\"\"\"&F.6#,&F0F1F1!\"\"F1F1\"\"#F5F1%\"hGF1/F0;F1%\"nG-F%6#-F)6$,&% \"aGF1*(,&*&F6F1F0F1F1F1F5F1-%!G6#*&,&%\"bGF1F@F5F1*&F6F1F:F1F5F1-FE6# *&,&FIF1F@F5F1F:F5F1F1/F0;F1F:" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 4 "Note" }{TEXT -1 41 ": The following examples use right sums. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 1 " }}{PARA 0 "" 0 "" {TEXT -1 15 " Evaluation of " }{XPPEDIT 18 0 "Int(x,x = 0 .. 1);" "6#-%$IntG6$%\"xG/F&;\"\"!\" \"\"" }{TEXT -1 33 " using the limit of a right sum. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 673 "Int(x,x= 0..1);\nit := %:\nfx := op(1,it): rg := op(2,it):\nxx := op(1,rg): aa \+ := op([2,1],rg): bb := op([2,2],rg):\nif not type(fx,polynom(realcons, xx)) then \nerror \"integrand must be a polynomial in %1\",xx end if: \nfn := unapply(fx,xx): i := 'i':\n``=Limit(Sum(``(fn(aa+i*((bb-aa)/n) ))*``((bb-aa)/n),i=1..n),n=infinity);\nff := expand(fn(aa+i*((bb-aa)/n ))*(bb-aa)/n):\ncfs := [coeffs(ff,i,'t')]: pwi := [t]:\n``=Limit(add(` `(cfs[j])*Sum(pwi[j],i=1..n),j=1..nops(cfs)),n=infinity);\nsms := map( _U->factor(sum(_U,i=1..n)),pwi):\n``=Limit(add(``(cfs[j])*``(sms[j]),j =1..nops(cfs)),n=infinity);\n``=Limit(add(cfs[j]*sms[j],j=1..nops(cfs) ),n=infinity);\n``=normal(rhs(%));\n``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$%\"xG/F&;\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$-%$SumG6$*&-F$6#*&%\"iG\"\"\"%\"nG!\"\"F 0-F$6#*&F0F0F1F2F0/F/;F0F1/F1%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$*&-F$6#*&\"\"\"F,*$)%\"nG\"\"#F,!\"\"F,-%$SumG6$% \"iG/F5;F,F/F,/F/%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-% &LimitG6$*&-F$6#*&\"\"\"F,*$)%\"nG\"\"#F,!\"\"F,-F$6#,$*(F0F1F/F,,&F/F ,F,F,F,F,F,/F/%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&Li mitG6$,$*(\"\"#!\"\"%\"nGF+,&F,\"\"\"F.F.F.F./F,%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$*(\"\"#!\"\"%\"nGF+,&F,\"\" \"F.F.F.F./F,%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\" \"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 265 47 "Check using the Fundamental Theorem of Calculus" }{TEXT -1 3 ": \+ " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 228 "Int(x,x=0..1);\nit := %: fx := op(1,it): rg := op(2,it):\nxx \+ := op(1,rg): aa := op([2,1],rg): bb := op([2,2],rg):\nF := unapply(int (fx,xx),xx): `anti-derivative: `, 'F(x)'=F(x);\n'F'(bb)-'F'(aa)=``(F( bb))-``(F(aa));\n``=F(bb)-F(aa);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-% $IntG6$%\"xG/F&;\"\"!\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3anti- derivative:~~G/-%\"FG6#%\"xG,$*&\"\"#!\"\"F(F+\"\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%\"FG6#\"\"\"F(-F&6#\"\"!!\"\",&-%!G6##F(\"\"#F( -F/F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\"\"\"\"#" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" } }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 2 " }}{PARA 0 "" 0 "" {TEXT -1 15 " Evaluation of " }{XPPEDIT 18 0 "Int(``(2-x^2),x = 0 .. 1 );" "6#-%$IntG6$-%!G6#,&\"\"#\"\"\"*$%\"xGF*!\"\"/F-;\"\"!F+" }{TEXT -1 33 " using the limit of a right sum. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 677 "Int(2-x^2,x=0..1);\n it := %: fx := op(1,it): rg := op(2,it):\nxx := op(1,rg): aa := op([2, 1],rg): bb := op([2,2],rg):\nif not type(fx,polynom(realcons,xx)) then \nerror \"integrand must be a polynomial in %1\",xx end if:\nfn := un apply(fx,xx): i := 'i':\n``=Limit(Sum(``(fn(aa+i*((bb-aa)/n)))*``((bb- aa)/n),i=1..n),n=infinity);\nff := expand(fn(aa+i*((bb-aa)/n))*(bb-aa) /n):\ncfs := [coeffs(ff,i,'t')]: pwi := [t]:\n``=Limit(add(``(cfs[j])* Sum(pwi[j],i=1..n),j=1..nops(cfs)),n=infinity);\nsms := map(_U->factor (sum(_U,i=1..n)),pwi):\n``=Limit(add(``(cfs[j])*``(sms[j]),j=1..nops(c fs)),n=infinity);\n``=Limit(add(cfs[j]*sms[j],j=1..nops(cfs)),n=infini ty);\n``=normal(rhs(%)); ``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&\"\"#\"\"\"*$)%\"xGF'F(!\"\"/F+;\"\"!F(" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$-%$SumG6$*&-F$6#,&\"\"# \"\"\"*&%\"iGF/%\"nG!\"#!\"\"F0-F$6#*&F0F0F3F5F0/F2;F0F3/F3%)infinityG " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,&*&-F$6#,$*&\"\"# \"\"\"%\"nG!\"\"F/F/-%$SumG6$F//%\"iG;F/F0F/F/*&-F$6#,$*&F/F/*$)F0\"\" $F/F1F1F/-F36$*$)F6F.F/F5F/F//F0%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,&*&-F$6#,$*&\"\"#\"\"\"%\"nG!\"\"F/F/-F $6#F0F/F/*&-F$6#,$*&F/F/*$)F0\"\"$F/F1F1F/-F$6#,$**\"\"'F1F0F/,&F0F/F/ F/F/,&*&F.F/F0F/F/F/F/F/F/F/F//F0%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,&\"\"#\"\"\"**\"\"'!\"\"%\"nG!\"#,&F.F* F*F*F*,&*&F)F*F.F*F*F*F*F*F-/F.%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$*(\"\"'!\"\",(*&\"#5\"\"\")%\"nG\"\"#F /F/*&\"\"$F/F1F/F+F/F+F/F1!\"#F//F1%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\"&\"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 267 47 "Check using the Fundamental Theorem of C alculus" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 234 "Int(2-x^2,x=0..1);\nit := %: fx := op(1,it): rg := op(2,it):\nxx := op(1,rg): aa := op([2,1],rg): bb := \+ op([2,2],rg):\nF := unapply(int(fx,xx),xx): `anti-derivative: `, 'F'( xx)=F(xx);\n'F'(bb)-'F'(aa)=``(F(bb))-``(F(aa));\n``=F(bb)-F(aa);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&\"\"#\"\"\"*$)%\"xGF'F(!\" \"/F+;\"\"!F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3anti-derivative:~~G /-%\"FG6#%\"xG,&*&\"\"#\"\"\"F(F,F,*&#F,\"\"$F,*$)F(F/F,F,!\"\"" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%\"FG6#\"\"\"F(-F&6#\"\"!!\"\",&-% !G6##\"\"&\"\"$F(-F/F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\"& \"\"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 3 \+ " }}{PARA 0 "" 0 "" {TEXT -1 15 " Evaluation of " }{XPPEDIT 18 0 "Int( ``(x^2+2*x+2),x = 1 .. 3);" "6#-%$IntG6$-%!G6#,(*$%\"xG\"\"#\"\"\"*&F, F-F+F-F-F,F-/F+;F-\"\"$" }{TEXT -1 33 " using the limit of a right sum . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 681 "Int(x^2+2*x+2,x=1..3);\nit := %: fx := op(1,it): rg \+ := op(2,it):\nxx := op(1,rg): aa := op([2,1],rg): bb := op([2,2],rg): \nif not type(fx,polynom(realcons,xx)) then \nerror \"integrand must b e a polynomial in %1\",xx end if:\nfn := unapply(fx,xx): i := 'i':\n`` =Limit(Sum(``(fn(aa+i*((bb-aa)/n)))*``((bb-aa)/n),i=1..n),n=infinity); \nff := expand(fn(aa+i*((bb-aa)/n))*(bb-aa)/n):\ncfs := [coeffs(ff,i,' t')]: pwi := [t]:\n``=Limit(add(``(cfs[j])*Sum(pwi[j],i=1..n),j=1..nop s(cfs)),n=infinity);\nsms := map(_U->factor(sum(_U,i=1..n)),pwi):\n``= Limit(add(``(cfs[j])*``(sms[j]),j=1..nops(cfs)),n=infinity);\n``=Limit (add(cfs[j]*sms[j],j=1..nops(cfs)),n=infinity);\n``=normal(rhs(%)); `` =value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,(*$)%\"xG \"\"#\"\"\"F+*&F*F+F)F+F+F*F+/F);F+\"\"$" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$-%$SumG6$*&-F$6#,(*$),&\"\"\"F2*(\"\"#F2%\"iGF2% \"nG!\"\"F2F4F2F2\"\"%F2*(F8F2F5F2F6F7F2F2-F$6#,$*&F4F2F6F7F2F2/F5;F2F 6/F6%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,(*& -F$6#,$*&\"#5\"\"\"%\"nG!\"\"F/F/-%$SumG6$F//%\"iG;F/F0F/F/*&-F$6#,$*& \"#;F/F0!\"#F/F/-F36$F6F5F/F/*&-F$6#,$*&\"\")F/F0!\"$F/F/-F36$*$)F6\" \"#F/F5F/F//F0%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&Li mitG6$,(*&-F$6#,$*&\"#5\"\"\"%\"nG!\"\"F/F/-F$6#F0F/F/*&-F$6#,$*&\"#;F /F0!\"#F/F/-F$6#,$*(\"\"#F1F0F/,&F0F/F/F/F/F/F/F/*&-F$6#,$*&\"\")F/F0! \"$F/F/-F$6#,$**\"\"'F1F0F/F@F/,&*&F?F/F0F/F/F/F/F/F/F/F//F0%)infinity G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,(\"#5\"\"\"*(\"\" )F*%\"nG!\"\",&F-F*F*F*F*F**,\"\"%F*\"\"$F.F-!\"#F/F*,&*&\"\"#F*F-F*F* F*F*F*F*/F-%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&Limit G6$,$**\"\"#\"\"\"\"\"$!\"\",(*&\"#JF+)%\"nGF*F+F+*&\"#=F+F2F+F+F*F+F+ F2!\"#F+/F2%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"#i\" \"$" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 47 " Check using the Fundamental Theorem of Calculus" }{TEXT -1 3 ": " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 238 "Int(x^2+2*x+2,x=1..3);\nit := %: fx := op(1,it): rg := op(2,it): \nxx := op(1,rg): aa := op([2,1],rg): bb := op([2,2],rg):\nF := unappl y(int(fx,xx),xx): `anti-derivative: `, 'F'(xx)=F(xx);\n'F'(bb)-'F'(aa )=``(F(bb))-``(F(aa));\n``=F(bb)-F(aa);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,(*$)%\"xG\"\"#\"\"\"F+*&F*F+F)F+F+F*F+/F);F+\"\"$" }} {PARA 11 "" 1 "" {XPPMATH 20 "6$%3anti-derivative:~~G/-%\"FG6#%\"xG,(* &#\"\"\"\"\"$F,*$)F(F-F,F,F,*$)F(\"\"#F,F,*&F2F,F(F,F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%\"FG6#\"\"$\"\"\"-F&6#F)!\"\",&-%!G6#\"#CF)-F /6##\"#5F(F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"#i\"\"$" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 4 " }}{PARA 0 " " 0 "" {TEXT -1 15 " Evaluation of " }{XPPEDIT 18 0 "Int(``(4*x-x^3),x = 0 .. 2);" "6#-%$IntG6$-%!G6#,&*&\"\"%\"\"\"%\"xGF,F,*$F-\"\"$!\"\"/ F-;\"\"!\"\"#" }{TEXT -1 33 " using the limit of a right sum. " }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 679 "Int(4*x-x^3,x=0..2);\nit := %: fx := op(1,it): rg := op(2,it):\nx x := op(1,rg): aa := op([2,1],rg): bb := op([2,2],rg):\nif not type(fx ,polynom(realcons,xx)) then \nerror \"integrand must be a polynomial i n %1\",xx end if:\nfn := unapply(fx,xx): i := 'i':\n``=Limit(Sum(``(fn (aa+i*((bb-aa)/n)))*``((bb-aa)/n),i=1..n),n=infinity);\nff := expand(f n(aa+i*((bb-aa)/n))*(bb-aa)/n):\ncfs := [coeffs(ff,i,'t')]: pwi := [t] :\n``=Limit(add(``(cfs[j])*Sum(pwi[j],i=1..n),j=1..nops(cfs)),n=infini ty);\nsms := map(_U->factor(sum(_U,i=1..n)),pwi):\n``=Limit(add(``(cfs [j])*``(sms[j]),j=1..nops(cfs)),n=infinity);\n``=Limit(add(cfs[j]*sms[ j],j=1..nops(cfs)),n=infinity);\n``=normal(rhs(%)); ``=value(rhs(%)); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*&\"\"%\"\"\"%\"xGF)F)* $)F*\"\"$F)!\"\"/F*;\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G -%&LimitG6$-%$SumG6$*&-F$6#,&*(\"\")\"\"\"%\"iGF1%\"nG!\"\"F1*(F0F1F2 \"\"$F3!\"$F4F1-F$6#,$*&\"\"#F1F3F4F1F1/F2;F1F3/F3%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,&*&-F$6#,$*&\"#;\"\"\" %\"nG!\"#F/F/-%$SumG6$%\"iG/F5;F/F0F/F/*&-F$6#,$*&F.F/F0!\"%!\"\"F/-F3 6$*$)F5\"\"$F/F6F/F//F0%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# /%!G-%&LimitG6$,&*&-F$6#,$*&\"#;\"\"\"%\"nG!\"#F/F/-F$6#,$*(\"\"#!\"\" F0F/,&F0F/F/F/F/F/F/F/*&-F$6#,$*&F.F/F0!\"%F7F/-F$6#,$*(\"\"%F7F0F6F8F 6F/F/F//F0%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG 6$,&*(\"\")\"\"\"%\"nG!\"\",&F,F+F+F+F+F+*(\"\"%F+F,!\"#F.\"\"#F-/F,%) infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$**\"\"% \"\"\",&%\"nGF+F+F+F+,&F-F+F+!\"\"F+F-!\"#F+/F-%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 268 47 "Check using the Fundamental Theorem of Calculus" }{TEXT -1 3 ": " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "Int(4*x-x^3,x=0..2);\nit := %: fx \+ := op(1,it): rg := op(2,it):\nxx := op(1,rg): aa := op([2,1],rg): bb : = op([2,2],rg):\nF := unapply(int(fx,xx),xx): `anti-derivative: `, 'F '(xx)=F(xx);\n'F'(bb)-'F'(aa)=``(F(bb))-``(F(aa));\n``=F(bb)-F(aa);" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*&\"\"%\"\"\"%\"xGF)F)*$)F *\"\"$F)!\"\"/F*;\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3anti- derivative:~~G/-%\"FG6#%\"xG,&*&\"\"#\"\"\")F(F+F,F,*&#F,\"\"%F,*$)F(F 0F,F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%\"FG6#\"\"#\"\"\"-F &6#\"\"!!\"\",&-%!G6#\"\"%F)-F0F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #/%!G\"\"%" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 10 "Example 5 " }}{PARA 0 "" 0 "" {TEXT -1 15 " Evaluation of " }{XPPEDIT 18 0 "Int(``(4*x-x^4),x = 0 .. 2);" "6# -%$IntG6$-%!G6#,&*&\"\"%\"\"\"%\"xGF,F,*$F-F+!\"\"/F-;\"\"!\"\"#" } {TEXT -1 33 " using the limit of a right sum. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 679 "Int(4*x-x^4 ,x=0..2);\nit := %: fx := op(1,it): rg := op(2,it):\nxx := op(1,rg): a a := op([2,1],rg): bb := op([2,2],rg):\nif not type(fx,polynom(realcon s,xx)) then \nerror \"integrand must be a polynomial in %1\",xx end if :\nfn := unapply(fx,xx): i := 'i':\n``=Limit(Sum(``(fn(aa+i*((bb-aa)/n )))*``((bb-aa)/n),i=1..n),n=infinity);\nff := expand(fn(aa+i*((bb-aa)/ n))*(bb-aa)/n):\ncfs := [coeffs(ff,i,'t')]: pwi := [t]:\n``=Limit(add( ``(cfs[j])*Sum(pwi[j],i=1..n),j=1..nops(cfs)),n=infinity);\nsms := map (_U->factor(sum(_U,i=1..n)),pwi):\n``=Limit(add(``(cfs[j])*``(sms[j]), j=1..nops(cfs)),n=infinity);\n``=Limit(add(cfs[j]*sms[j],j=1..nops(cfs )),n=infinity);\n``=normal(rhs(%)); ``=value(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$,&*&\"\"%\"\"\"%\"xGF)F)*$)F*F(F)!\"\"/F* ;\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$-%$SumG6 $*&-F$6#,&*(\"\")\"\"\"%\"iGF1%\"nG!\"\"F1*(\"#;F1F2\"\"%F3!\"%F4F1-F$ 6#,$*&\"\"#F1F3F4F1F1/F2;F1F3/F3%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,&*&-F$6#,$*&\"#;\"\"\"%\"nG!\"#F/F/-%$S umG6$%\"iG/F5;F/F0F/F/*&-F$6#,$*&\"#KF/F0!\"&!\"\"F/-F36$*$)F5\"\"%F/F 6F/F//F0%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$ ,&*&-F$6#,$*&\"#;\"\"\"%\"nG!\"#F/F/-F$6#,$*(\"\"#!\"\"F0F/,&F0F/F/F/F /F/F/F/*&-F$6#,$*&\"#KF/F0!\"&F7F/-F$6#,$*,\"#IF7F0F/,&*&F6F/F0F/F/F/F /F/F8F/,(*&\"\"$F/)F0F6F/F/*&FIF/F0F/F/F/F7F/F/F/F//F0%)infinityG" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,&*(\"\")\"\"\"%\"nG!\" \",&F,F+F+F+F+F+*.\"#;F+\"#:F-F,!\"%,&*&\"\"#F+F,F+F+F+F+F+F.F+,(*&\" \"$F+)F,F5F+F+*&F8F+F,F+F+F+F-F+F-/F,%)infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G-%&LimitG6$,$*,\"\")\"\"\"\"#:!\"\",&%\"nGF+F+F+F+, **&\"\"$F+)F/F2F+F+*&\"#=F+)F/\"\"#F+F-*&F7F+F/F+F-F7F+F+F/!\"%F+/F/%) infinityG" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\")\"\"&" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 47 "Check us ing the Fundamental Theorem of Calculus" }{TEXT -1 3 ": " }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 236 "Int( 4*x-x^4,x=0..2);\nit := %: fx := op(1,it): rg := op(2,it):\nxx := op(1 ,rg): aa := op([2,1],rg): bb := op([2,2],rg):\nF := unapply(int(fx,xx) ,xx): `anti-derivative: `, 'F'(xx)=F(xx);\n'F'(bb)-'F'(aa)=``(F(bb))- ``(F(aa));\n``=F(bb)-F(aa);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG 6$,&*&\"\"%\"\"\"%\"xGF)F)*$)F*F(F)!\"\"/F*;\"\"!\"\"#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%3anti-derivative:~~G/-%\"FG6#%\"xG,&*&\"\"#\"\"\" )F(F+F,F,*&#F,\"\"&F,*$)F(F0F,F,!\"\"" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%\"FG6#\"\"#\"\"\"-F&6#\"\"!!\"\",&-%!G6##\"\")\"\"&F)-F0F+F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%!G#\"\")\"\"&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1 ";" }}}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "4 0 0" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }