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" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 256 5 "Maple" }{TEXT -1 135 " will help with graphing, with visualizing the subdivi sion of the solid into approximating boxes, and with evaluating sums a nd limits." }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 5 "" 0 " " {TEXT 273 15 "Solved Example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 50 "Find the volume of the solid between the \+ surface " }{TEXT 257 1 "z" }{TEXT -1 7 " = 4 - " }{TEXT 258 3 "x y" } {TEXT -1 31 " and the rectangle (square), " }{TEXT 259 1 "R" }{TEXT -1 24 " = [0, 2] x [-1, 1]." }}}{PARA 5 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 1 " \+ " }{TEXT 270 0 "" }{TEXT 271 9 "Solution:" }}{PARA 0 "" 0 "" {TEXT -1 52 "We will start with plotting the surface in question:" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "with(plots):" }}{PARA 7 "" 1 "" {TEXT -1 50 "Warning, the name changecoords has been redefined\n" }}} {EXCHG {PARA 0 "> " 0 "" 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The height of a \+ box is the value of the function at a point in the subrectangle." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "dx:=2/n;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dxG,$*&\"\"\"F'%\"nG !\"\"\"\"#" }}}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "dy:=2/m;" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#dyG,$*&\"\"\"F'%\"mG!\"\"\"\"#" }}} {PARA 11 "" 1 "" {TEXT -1 0 "" }}{PARA 11 "" 1 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "V:=Sum(Sum(dx*dy*f(i*dx,1-j* dy),i=1..n),j=0..m-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG-%$Sum G6$-F&6$,$*&,&\"\"%\"\"\"*&*(\"\"#F.,&F.F.*&*&F1F.%\"jGF.F.%\"mG!\"\"F 7F.%\"iGF.F.%\"nGF7F7F.*&F9F.F6F.F7F-/F8;F.F9/F5;\"\"!,&F6F.F.F7" }}} {PARA 11 "" 1 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "volume:=evalf(value(subs(n=6,m=6,V)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'volumeG$\"+AAAA:!\")" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 102 "This is the approximation of the vo lume of the solid by 36 boxes. Let's look at the plot of the boxes." } }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "with(linalg):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "matrixplot(matrix(6,6,(j,k)->4-j/3* (1-(k-1)/3)),heights=histogram, axes=boxed);" }}{PARA 13 "" 1 "" {GLPLOT3D 328 207 207 {PLOTDATA 3 "6%-%)POLYGONSG6dx7&7%\"\"\"F(\"\"!7 %$\"+AAs**>!\"*F(F)7%F+F+F)7%F(F+F)7&7%F(F(#\"#6\"\"$7%F+F(F27%F+F+F27 %F(F+F27&F'F*F5F17&F*F.F6F57&F.F/F7F67&F'F/F7F17&7%\"\"#F(F)7%$\"+AAs* *HF-F(F)7%F@F+F)7%F>F+F)7&7%F>F(#\"#5F47%F@F(FF7%F@F+FF7%F>F+FF7&F=F?F HFE7&F?FBFIFH7&FBFCFJFI7&F=FCFJFE7&7%F4F(F)7%$\"+AAs**RF-F(F)7%FRF+F)7 %F4F+F)7&7%F4F(F47%FRF(F47%FRF+F47%F4F+F47&FPFQFXFW7&FQFTFYFX7&FTFUFZF Y7&FPFUFZFW7&7%\"\"%F(F)7%$\"+AAs**\\F-F(F)7%F]oF+F)7%F[oF+F)7&7%F[oF( #\"\")F47%F]oF(Fco7%F]oF+Fco7%F[oF+Fco7&FjnF\\oFeoFbo7&F\\oF_oFfoFeo7& F_oF`oFgoFfo7&FjnF`oFgoFbo7&7%\"\"&F(F)7%$\"+AAs**fF-F(F)7%F`pF+F)7%F^ pF+F)7&7%F^pF(#\"\"(F47%F`pF(Ffp7%F`pF+Ffp7%F^pF+Ffp7&F]pF_pFhpFep7&F_ 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0 0 "Curve 1" }}}}{PARA 0 "" 0 "" {TEXT -1 58 "Let's now app roximate the volume with 20 boxes along the " }{TEXT 274 1 "x" } {TEXT -1 30 "-axis and 30 boxes along the " }{TEXT 275 1 "y" }{TEXT 276 1 "-" }{TEXT -1 5 "axis." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "volume:=evalf(value(subs(n=2 0,m=30,V)));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%'volumeG$\"++++'e\"! \")" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "matrixplot(matrix(20 ,30,(j,k)->4-j/10*(1-(k-1)/15)),heights=histogram,axes=boxed);" }} {PARA 13 "" 1 "" {GLPLOT3D 400 300 300 {PLOTDATA 3 "6%-%)POLYGONSG6\\ \\y7&7%\"\"\"F(\"\"!7%$\"+LL)***>!\"*F(F)7%F+F+F)7%F(F+F)7&7%F(F(#\"#R \"#57%F+F(F27%F+F+F27%F(F+F27&F'F*F5F17&F*F.F6F57&F.F/F7F67&F'F/F7F17& 7%\"\"#F(F)7%$\"+LL)***HF-F(F)7%F@F+F)7%F>F+F)7&7%F>F(#\"#>\"\"&7%F@F( FF7%F@F+FF7%F>F+FF7&F=F?FIFE7&F?FBFJFI7&FBFCFKFJ7&F=FCFKFE7&7%\"\"$F(F )7%$\"+LL)***RF-F(F)7%FTF+F)7%FRF+F)7&7%FRF(#\"#PF47%FTF(FZ7%FTF+FZ7%F 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