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4 2 0 2 0 2 2 -1 1 }} {SECT 0 {PARA 242 "" 0 "" {TEXT 260 14 "Computer Lab 4" }}{PARA 243 "" 0 "" {TEXT 260 18 "Iterated Integrals" }}{PARA 202 "" 0 "" {TEXT 294 0 "" }}{PARA 202 "" 0 "" {TEXT 268 87 "In this lab you will learn how \+ to calculate iterated and double integrals using MAPLE. " }}{PARA 202 "" 0 "" {TEXT 294 0 "" }}{SECT 0 {PARA 238 "" 0 "" {TEXT 252 8 "Exampl es" }}{SECT 1 {PARA 213 "" 0 "" {TEXT 246 49 " Example I: Iterated int egrals, Leftsum, Rightsum" }}{PARA 202 "" 0 "" {TEXT 294 0 "" }}{PARA 202 "" 0 "" {TEXT 268 14 "(a) Plot " }{XPPEDIT 18 0 "f(x, y) = si n(x+y);" "6#/-%\"fG6$%\"xG%\"yG-%$sinG6#,&F'\"\"\"F(F-" }{TEXT 268 1 " " }{TEXT 268 1 " " }{TEXT 268 1 " " }{TEXT 268 14 " for R = \{(" } {TEXT 249 1 "x" }{TEXT 268 1 "," }{TEXT 249 1 "y" }{TEXT 268 6 ") | 0 " }{TEXT 209 1 "<" }{TEXT 268 1 " " }{TEXT 249 1 "x" }{TEXT 268 1 " " }{TEXT 209 1 "<" }{TEXT 268 1 " " }{XPPEDIT 213 0 "Pi;" "6#I#PiG%*prot ectedG" }{TEXT 268 1 " " }{TEXT 268 6 "/2, 0" }{TEXT 209 1 "<" }{TEXT 268 1 " " }{TEXT 249 1 "y" }{TEXT 268 1 " " }{TEXT 209 1 "<" }{TEXT 268 1 " " }{XPPEDIT 213 0 "Pi;" "6#I#PiG%*protectedG" }{TEXT 268 1 " " }{TEXT 268 57 "/4 \}. Find the exact volume using an iterated integr al." }}{PARA 202 "" 0 "" {TEXT 294 0 "" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 218 36 "restart: with(plots): with(student);" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 218 48 "plot3d(sin(x+y),x=0..Pi/2,y=0.. Pi/4,axes=boxed);" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 218 54 "V: =Int(Int(sin(x+y),x=0..Pi/2),y=0..Pi/4): V=value(V);" }}}{PARA 202 "" 0 "" {TEXT 294 0 "" }}{PARA 202 "" 0 "" {TEXT 268 114 "(b) Approxima te the above integral with left and right Riemann double sums. Use m \+ = 6 and n = 10 subdivisions." }}{PARA 202 "" 0 "" {TEXT 294 0 "" }} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 218 79 "Vleft:=leftsum(leftsum(s in(x+y),x=0..Pi/2,6),y=0..Pi/4,10): Vleft=evalf(Vleft);" }}}{PARA 202 "" 0 "" {TEXT 294 0 "" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 218 84 "Vright:=rightsum(rightsum(sin(x+y),x=0..Pi/2,6),y=0..Pi/4,10): Vright =evalf(Vright);" }}}{PARA 202 "" 0 "" {TEXT 294 0 "" }}}{PARA 202 "" 0 "" {TEXT 294 0 "" }}{SECT 1 {PARA 213 "" 0 "" {TEXT 246 21 " Example \+ 2: Doubleint" }}{PARA 202 "" 0 "" {TEXT 268 20 "Find the volume of " }{XPPEDIT 18 0 "f(x, y) = sin(x+y);" "6#/-%\"fG6$%\"xG%\"yG-%$sinG6#,& F'\"\"\"F(F-" }{TEXT 268 1 " " }{TEXT 268 1 " " }{TEXT 268 1 " " } {TEXT 268 14 " for R = \{(" }{TEXT 249 1 "x" }{TEXT 268 1 "," }{TEXT 249 1 "y" }{TEXT 268 6 ") | 0" }{TEXT 209 1 "<" }{TEXT 268 1 " " } {TEXT 249 1 "x" }{TEXT 268 1 " " }{TEXT 209 1 "<" }{TEXT 268 1 " " } {XPPEDIT 213 0 "Pi;" "6#I#PiG%*protectedG" }{TEXT 268 1 " " }{TEXT 268 6 "/2, 0" }{TEXT 209 1 "<" }{TEXT 268 1 " " }{TEXT 249 1 "y" } {TEXT 268 1 " " }{TEXT 209 1 "<" }{TEXT 268 1 " " }{XPPEDIT 213 0 "Pi; " "6#I#PiG%*protectedG" }{TEXT 268 1 " " }{TEXT 268 40 "/4 \}using the Maple function Doubleint." }}{PARA 202 "" 0 "" {TEXT 294 0 "" }} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 218 55 "V:=Doubleint(sin(x+y),x= 0..Pi/2,y=0..Pi/4): V=value(V);" }}}{PARA 202 "" 0 "" {TEXT 294 0 "" } }}{PARA 227 "" 0 "" {TEXT 295 0 "" }}{SECT 1 {PARA 213 "" 0 "" {TEXT 246 1 " " }{TEXT 246 47 "Example 3: Double integral over general regi on" }}{PARA 202 "" 0 "" {TEXT 268 43 "Consider the solid between the p araboloids " }{XPPEDIT 18 0 "z = 2*x^2+y^2;" "6#/%\"zG,&*&\"\"#\"\"\") %\"xGF'F(F(*$)%\"yGF'F(F(" }{TEXT 268 1 " " }{TEXT 268 1 " " }{TEXT 268 1 " " }{TEXT 268 5 "and " }{XPPEDIT 18 0 "z = 8-x^2-2*y^2;" "6#/% \"zG,(\"\")\"\"\"*$)%\"xG\"\"#F'!\"\"*&F+F')%\"yGF+F'F," }{TEXT 268 1 " " }{TEXT 268 1 " " }{TEXT 268 2 " ," }{TEXT 268 27 " and inside the \+ cylinder " }{XPPEDIT 18 0 "x^2+y^2 = 1;" "6#/,&*$)%\"xG\"\"#\"\"\"F) *$)%\"yGF(F)F)F)" }{TEXT 268 1 " " }{TEXT 268 1 " " }{TEXT 268 1 "." } }{PARA 202 "" 0 "" {TEXT 294 0 "" }}{PARA 202 "" 0 "" {TEXT 268 51 "a) Plot the paraboloids on the appropriate domain. " }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 218 13 "f:=2*x^2+y^2;" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 218 15 "g:=8-x^2-2*y^2;" }}}{PARA 202 "" 0 "" {TEXT 294 0 "" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 218 64 "plot3d(\{f,g \},x=-1..1,y=-sqrt(1-x^2)..sqrt(1-x^2),axes=normal);" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 215 7 "NOTE: " }{TEXT 268 41 "The domain of integra tion is the circle " }{XPPEDIT 18 0 "x^2+y^2 = 1;" "6#/,&*$)%\"xG\"\" #\"\"\"F)*$)%\"yGF(F)F)F)" }{TEXT 268 1 " " }{TEXT 268 1 " " }{TEXT 268 1 " " }{TEXT 268 27 " which can be described by " }{XPPEDIT 2 0 "( -1 <= x) <= 1;" "6#11,$\"\"\"!\"\"I\"xG6\"F&" }{TEXT 268 1 " " }{TEXT 268 1 " " }{TEXT 268 1 " " }{TEXT 268 5 "and " }{TEXT 268 1 " " } {XPPEDIT 2 0 "(-sqrt(1-x^2) <= y) <= sqrt(1-x^2);" "6#11,$-I%sqrtG6$%* protectedGI(_syslibG6\"6#,&\"\"\"F.*$)I\"xGF+\"\"#F.!\"\"F3I\"yGF+F&" }{TEXT 268 1 " " }{TEXT 268 1 " " }{TEXT 268 6 "; . " }}{PARA 202 "" 0 "" {TEXT 268 19 "Also note that for " }{TEXT 249 1 "x" }{TEXT 268 5 " and " }{TEXT 249 1 "y" }{TEXT 268 35 " inside this domain the para boloid " }{TEXT 249 1 "g" }{TEXT 268 10 " is above " }{TEXT 249 1 "f" }{TEXT 268 1 "." }}{PARA 202 "" 0 "" {TEXT 294 0 "" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 268 84 "We can plot the surfaces and the domain of i ntegration using the following commands:" }}}{EXCHG {PARA 234 "> " 0 " " {MPLTEXT 1 207 16 "with(plottools):" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 218 66 "q:=plot3d(\{f,g\},x=-1..1,y=-sqrt(1-x^2)..sqrt(1-x^ 2),axes=boxed):" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 218 55 "p:=i mplicitplot(x^2+y^2=1,x=-1..1,y=-1..1,thickness=4):" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 218 29 "h:=transform((x,y)->[x,y,0]):" }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 218 20 "display(\{q,h(p)\});" }} }{PARA 233 "" 0 "" {TEXT 296 0 "" }}{PARA 202 "" 0 "" {TEXT 294 0 "" } }{PARA 202 "" 0 "" {TEXT 268 57 "b) Find the volume of the solid usin g a Double Integral." }}{PARA 202 "" 0 "" {TEXT 294 0 "" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 218 66 "V:=Doubleint(g-f,y=-sqrt(1-x^2) ..sqrt(1-x^2),x=-1..1): V=value(V);" }}}{PARA 202 "" 0 "" {TEXT 294 0 "" }}{PARA 202 "" 0 "" {TEXT 294 0 "" }}{PARA 229 "" 0 "" {TEXT 297 0 "" }}}{PARA 202 "" 0 "" {TEXT 294 0 "" }}}{PARA 202 "" 0 "" {TEXT 294 0 "" }}{SECT 1 {PARA 212 "" 0 "" {TEXT 298 1 " " }}}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }