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" }}{PARA 202 "" 0 "" {TEXT 293 0 "" }} {SECT 1 {PARA 237 "" 0 "" {TEXT 291 8 "Examples" }}{SECT 0 {PARA 212 " " 0 "" {TEXT 279 49 " Example I: Iterated integrals, Leftsum, Rightsum " }}{PARA 202 "" 0 "" {TEXT 293 0 "" }}{PARA 202 "" 0 "" {TEXT 239 14 "(a) Plot " }{XPPEDIT 18 0 "f(x, y) = sin(x+y);" "6#/-%\"fG6$%\"x G%\"yG-%$sinG6#,&F'\"\"\"F(F-" }{TEXT 239 1 " " }{TEXT 239 1 " " } {TEXT 239 1 " " }{TEXT 239 14 " for R = \{(" }{TEXT 282 1 "x" }{TEXT 239 1 "," }{TEXT 282 1 "y" }{TEXT 239 6 ") | 0" }{TEXT 211 1 "<" } {TEXT 239 1 " " }{TEXT 282 1 "x" }{TEXT 239 1 " " }{TEXT 211 1 "<" } {TEXT 239 1 " " }{XPPEDIT 220 0 "Pi;" "6#I#PiG%*protectedG" }{TEXT 239 1 " " }{TEXT 239 6 "/2, 0" }{TEXT 211 1 "<" }{TEXT 239 1 " " } {TEXT 282 1 "y" }{TEXT 239 1 " " }{TEXT 211 1 "<" }{TEXT 239 1 " " } {XPPEDIT 220 0 "Pi;" "6#I#PiG%*protectedG" }{TEXT 239 1 " " }{TEXT 239 57 "/4 \}. Find the exact volume using an iterated integral." }} {PARA 202 "" 0 "" {TEXT 293 0 "" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 36 "restart: with(plots): with(student);" }}}{EXCHG {PARA 202 " > " 0 "" {MPLTEXT 1 227 48 "plot3d(sin(x+y),x=0..Pi/2,y=0..Pi/4,axes=b oxed);" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 54 "V:=Int(Int(si n(x+y),x=0..Pi/2),y=0..Pi/4): V=value(V);" }}}{PARA 202 "" 0 "" {TEXT 293 0 "" }}{PARA 202 "" 0 "" {TEXT 239 114 "(b) Approximate the abov e integral with left and right Riemann double sums. Use m = 6 and n \+ = 10 subdivisions." }}{PARA 202 "" 0 "" {TEXT 293 0 "" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 79 "Vleft:=leftsum(leftsum(sin(x+y),x=0.. Pi/2,6),y=0..Pi/4,10): Vleft=evalf(Vleft);" }}}{PARA 202 "" 0 "" {TEXT 293 0 "" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 84 "Vright:=righ tsum(rightsum(sin(x+y),x=0..Pi/2,6),y=0..Pi/4,10): Vright=evalf(Vright );" }}}{PARA 202 "" 0 "" {TEXT 293 0 "" }}}{PARA 202 "" 0 "" {TEXT 293 0 "" }}{SECT 0 {PARA 212 "" 0 "" {TEXT 279 21 " Example 2: Doublei nt" }}{PARA 202 "" 0 "" {TEXT 239 20 "Find the volume of " }{XPPEDIT 18 0 "f(x, y) = sin(x+y);" "6#/-%\"fG6$%\"xG%\"yG-%$sinG6#,&F'\"\"\"F( F-" }{TEXT 239 1 " " }{TEXT 239 1 " " }{TEXT 239 1 " " }{TEXT 239 14 " for R = \{(" }{TEXT 282 1 "x" }{TEXT 239 1 "," }{TEXT 282 1 "y" } {TEXT 239 6 ") | 0" }{TEXT 211 1 "<" }{TEXT 239 1 " " }{TEXT 282 1 "x " }{TEXT 239 1 " " }{TEXT 211 1 "<" }{TEXT 239 1 " " }{XPPEDIT 220 0 " Pi;" "6#I#PiG%*protectedG" }{TEXT 239 1 " " }{TEXT 239 6 "/2, 0" } {TEXT 211 1 "<" }{TEXT 239 1 " " }{TEXT 282 1 "y" }{TEXT 239 1 " " } {TEXT 211 1 "<" }{TEXT 239 1 " " }{XPPEDIT 220 0 "Pi;" "6#I#PiG%*prote ctedG" }{TEXT 239 1 " " }{TEXT 239 40 "/4 \}using the Maple function D oubleint." }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 0 "Typesetting:-mrow (Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"with\"), T ypesetting:-mo(\"(\", form = \"prefix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"thinmathspace\", st retchy = \"true\", symmetric = \"false\", maxsize = \"infinity\", mins ize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", font_style_name = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\"), Typesetting:-mi(\"plo ts\"), Typesetting:-mo(\")\", form = \"postfix\", fence = \"true\", se parator = \"false\", lspace = \"thinmathspace\", rspace = \"verythinma thspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"inf inity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", font_style_name = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\"), Typesett ing:-mo(\":\", form = \"infix\", fence = \"false\", separator = \"fals e\", lspace = \"thickmathspace\", rspace = \"thickmathspace\", stretch y = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize \+ = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"fa lse\", font_style_name = \"2D Input\", size = \"12\", foreground = \"[ 255,0,0]\", background = \"[255,255,255]\"), Typesetting:-mspace(heigh t = \"0.0 ex\", width = \"0.5 em\", depth = \"0.0 ex\", linebreak = \" auto\"), Typesetting:-mi(\"with\"), Typesetting:-mo(\"(\", form = \"pr efix\", fence = \"true\", separator = \"false\", lspace = \"thinmathsp ace\", rspace = \"thinmathspace\", stretchy = \"true\", symmetric = \" false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_style_name = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", background = \"[2 55,255,255]\"), Typesetting:-mi(\"student\"), Typesetting:-mo(\")\", f orm = \"postfix\", fence = \"true\", separator = \"false\", lspace = \+ \"thinmathspace\", rspace = \"verythinmathspace\", stretchy = \"true\" , symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", larg eop = \"false\", movablelimits = \"false\", accent = \"false\", font_s tyle_name = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", b ackground = \"[255,255,255]\"), Typesetting:-mo(\";\", form = \"infix \", fence = \"false\", separator = \"true\", lspace = \"0em\", rspace \+ = \"thickmathspace\", stretchy = \"false\", symmetric = \"false\", max size = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimi ts = \"false\", accent = \"false\", font_style_name = \"2D Input\", si ze = \"12\", foreground = \"[255,0,0]\", background = \"[255,255,255] \")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Typesetti ngGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6--F,6#Q%withF'-I#moGF$63Q\"(F'/%%fo rmGQ'prefixF'/%&fenceGQ%trueF'/%*separatorGQ&falseF'/%'lspaceGQ.thinma thspaceF'/%'rspaceGFC/%)stretchyGF=/%*symmetricGF@/%(maxsizeGQ)infinit yF'/%(minsizeGQ\"1F'/%(largeopGF@/%.movablelimitsGF@/%'accentGF@/%0fon t_style_nameGQ)2D~InputF'/%%sizeGQ#12F'/%+foregroundGQ*[255,0,0]F'/%+b ackgroundGQ.[255,255,255]F'-F,6#Q&plotsF'-F563Q\")F'/F9Q(postfixF'F;F> FA/FEQ2verythinmathspaceF'FFFHFJFMFPFRFTFVFYFfnFin-F563Q\":F'/F9Q&infi xF'/F/FBQ/thickmathspaceF'/FEF]p/FGF@FHFJFMFPFRFTFVFYFfnFin-I'msp aceGF$6&/%'heightGQ'0.0~exF'/%&widthGQ'0.5~emF'/%&depthGFep/%*linebrea kGQ%autoF'F1F4-F,6#Q(studentF'F_o-F563Q\";F'FioF[p/F?F=/FBQ$0emF'F^pF_ pFHFJFMFPFRFTFVFYFfnFinF+" }}}{PARA 202 "" 0 "" {TEXT 293 0 "" }} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 55 "V:=Doubleint(sin(x+y),x= 0..Pi/2,y=0..Pi/4): V=value(V);" }}}{PARA 202 "" 0 "" {TEXT 293 0 "" } }}{PARA 224 "" 0 "" {TEXT 294 0 "" }}{SECT 0 {PARA 212 "" 0 "" {TEXT 279 1 " " }{TEXT 279 47 "Example 3: Double integral over general regi on" }}{EXCHG {PARA 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\"infix\", fence = \"false\", separator = \"false\", \+ lspace = \"thickmathspace\", rspace = \"thickmathspace\", stretchy = \+ \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \" 1\", largeop = \"false\", movablelimits = \"false\", accent = \"false \", font_style_name = \"2D Input\", size = \"12\", foreground = \"[255 ,0,0]\", background = \"[255,255,255]\"), Typesetting:-mspace(height = \"0.0 ex\", width = \"0.5 em\", depth = \"0.0 ex\", linebreak = \"aut o\"), Typesetting:-mi(\"with\"), Typesetting:-mo(\"(\", form = \"prefi x\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace \", rspace = \"thinmathspace\", stretchy = \"true\", symmetric = \"fal se\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", font_style_name = \"2D In put\", size = \"12\", foreground = \"[255,0,0]\", background = \"[255, 255,255]\"), Typesetting:-mi(\"student\"), Typesetting:-mo(\")\", form = \"postfix\", fence = \"true\", separator = \"false\", lspace = \"th inmathspace\", rspace = \"verythinmathspace\", stretchy = \"true\", sy mmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop \+ = \"false\", movablelimits = \"false\", accent = \"false\", font_style _name = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", backg round = \"[255,255,255]\"), Typesetting:-mo(\";\", form = \"infix\", f ence = \"false\", separator = \"true\", lspace = \"0em\", rspace = \"t hickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize \+ = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \+ \"false\", accent = \"false\", font_style_name = \"2D Input\", size = \+ \"12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\")), \+ Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_ syslibGF'6%-I#miGF$6#Q!F'-F#6--F,6#Q%withF'-I#moGF$63Q\"(F'/%%formGQ'p refixF'/%&fenceGQ%trueF'/%*separatorGQ&falseF'/%'lspaceGQ.thinmathspac eF'/%'rspaceGFC/%)stretchyGF=/%*symmetricGF@/%(maxsizeGQ)infinityF'/%( minsizeGQ\"1F'/%(largeopGF@/%.movablelimitsGF@/%'accentGF@/%0font_styl e_nameGQ)2D~InputF'/%%sizeGQ#12F'/%+foregroundGQ*[255,0,0]F'/%+backgro undGQ.[255,255,255]F'-F,6#Q&plotsF'-F563Q\")F'/F9Q(postfixF'F;F>FA/FEQ 2verythinmathspaceF'FFFHFJFMFPFRFTFVFYFfnFin-F563Q\":F'/F9Q&infixF'/F< F@F>/FBQ/thickmathspaceF'/FEF]p/FGF@FHFJFMFPFRFTFVFYFfnFin-I'mspaceGF$ 6&/%'heightGQ'0.0~exF'/%&widthGQ'0.5~emF'/%&depthGFep/%*linebreakGQ%au toF'F1F4-F,6#Q(studentF'F_o-F563Q\";F'FioF[p/F?F=/FBQ$0emF'F^pF_pFHFJF MFPFRFTFVFYFfnFinF+" }}}{PARA 202 "" 0 "" {TEXT 239 43 "Consider the s olid between the paraboloids " }{XPPEDIT 18 0 "z = 2*x^2+y^2;" "6#/%\" zG,&*&\"\"#\"\"\")%\"xGF'F(F(*$)%\"yGF'F(F(" }{TEXT 239 1 " " }{TEXT 239 1 " " }{TEXT 239 1 " " }{TEXT 239 5 "and " }{XPPEDIT 18 0 "z = 8- x^2-2*y^2;" "6#/%\"zG,(\"\")\"\"\"*$)%\"xG\"\"#F'!\"\"*&F+F')%\"yGF+F' F," }{TEXT 239 1 " " }{TEXT 239 1 " " }{TEXT 239 2 " ," }{TEXT 239 27 " and inside the cylinder " }{XPPEDIT 18 0 "x^2+y^2 = 1;" "6#/,&*$)% \"xG\"\"#\"\"\"F)*$)%\"yGF(F)F)F)" }{TEXT 239 1 " " }{TEXT 239 1 " " } {TEXT 239 1 "." }}{PARA 202 "" 0 "" {TEXT 293 0 "" }}{PARA 202 "" 0 "" {TEXT 239 51 "a) Plot the paraboloids on the appropriate domain. " }} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 13 "f:=2*x^2+y^2;" }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 15 "g:=8-x^2-2*y^2;" }}} {PARA 202 "" 0 "" {TEXT 293 0 "" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 64 "plot3d(\{f,g\},x=-1..1,y=-sqrt(1-x^2)..sqrt(1-x^2),axes=nor mal);" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 222 7 "NOTE: " }{TEXT 239 41 "The domain of integration is the circle " }{XPPEDIT 18 0 "x^2+y^2 = 1;" "6#/,&*$)%\"xG\"\"#\"\"\"F)*$)%\"yGF(F)F)F)" }{TEXT 239 1 " " } {TEXT 239 1 " " }{TEXT 239 1 " " }{TEXT 239 27 " which can be describe d by " }{XPPEDIT 2 0 "(-1 <= x) <= 1;" "6#11,$\"\"\"!\"\"I\"xG6\"F&" } {TEXT 239 1 " " }{TEXT 239 1 " " }{TEXT 239 1 " " }{TEXT 239 5 "and " }{TEXT 239 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 239 1 " " }{TEXT 239 1 " " }{TEXT 239 6 "; \+ . " }}{PARA 202 "" 0 "" {TEXT 239 19 "Also note that for " }{TEXT 282 1 "x" }{TEXT 239 5 " and " }{TEXT 282 1 "y" }{TEXT 239 35 " inside this domain the paraboloid " }{TEXT 282 1 "g" }{TEXT 239 10 " is abov e " }{TEXT 282 1 "f" }{TEXT 239 1 "." }}{PARA 202 "" 0 "" {TEXT 293 0 "" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 239 84 "We can plot the surfaces \+ and the domain of integration using the following commands:" }}} {EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 209 16 "with(plottools):" }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 66 "q:=plot3d(\{f,g\},x=-1.. 1,y=-sqrt(1-x^2)..sqrt(1-x^2),axes=boxed):" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 55 "p:=implicitplot(x^2+y^2=1,x=-1..1,y=-1..1,thick ness=4):" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 29 "h:=transfor m((x,y)->[x,y,0]):" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 20 "d isplay(\{q,h(p)\});" }}}{PARA 232 "" 0 "" {TEXT 295 0 "" }}{PARA 202 " " 0 "" {TEXT 293 0 "" }}{PARA 202 "" 0 "" {TEXT 239 57 "b) Find the v olume of the solid using a Double Integral." }}{PARA 202 "" 0 "" {TEXT 293 0 "" }}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 66 "V:=Doubleint (g-f,y=-sqrt(1-x^2)..sqrt(1-x^2),x=-1..1): V=value(V);" }}}{PARA 202 " " 0 "" {TEXT 293 0 "" }}{PARA 202 "" 0 "" {TEXT 293 0 "" }}{PARA 228 " " 0 "" {TEXT 296 0 "" }}}{PARA 202 "" 0 "" {TEXT 293 0 "" }}}{PARA 202 "" 0 "" {TEXT 293 0 "" }}{SECT 1 {PARA 211 "" 0 "" {TEXT 297 51 " Exe rcises Solve at least one problem and turn in. " }}{EXCHG {PARA 0 "> " 0 "" {XPPEDIT 19 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typesett ing:-mrow(Typesetting:-mi(\"with\"), Typesetting:-mo(\"(\", form = \"p refix\", fence = \"true\", separator = \"false\", lspace = \"thinmaths pace\", rspace = \"thinmathspace\", stretchy = \"true\", symmetric = \+ \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mi(\"plots\"), Typesetting:-mo(\")\", form = \"postfix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"verythinmathspace\", stretchy = \"true \", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", la rgeop = \"false\", movablelimits = \"false\", accent = \"false\", font _style_name = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\"), Typesetting:-mo(\":\", form = \"infi x\", fence = \"false\", separator = \"false\", lspace = \"thickmathspa ce\", rspace = \"thickmathspace\", stretchy = \"false\", symmetric = \+ \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false \", movablelimits = \"false\", accent = \"false\", font_style_name = \+ \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", background = \+ \"[255,255,255]\"), Typesetting:-mspace(height = \"0.0 ex\", width = \+ \"0.5 em\", depth = \"0.0 ex\", linebreak = \"auto\"), Typesetting:-mi (\"with\"), Typesetting:-mo(\"(\", form = \"prefix\", fence = \"true\" , separator = \"false\", lspace = \"thinmathspace\", rspace = \"thinma thspace\", stretchy = \"true\", symmetric = \"false\", maxsize = \"inf inity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false \", accent = \"false\", font_style_name = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\"), Typesett ing:-mi(\"student\"), Typesetting:-mo(\")\", form = \"postfix\", fence = \"true\", separator = \"false\", lspace = \"thinmathspace\", rspace = \"verythinmathspace\", stretchy = \"true\", symmetric = \"false\", \+ maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablel imits = \"false\", accent = \"false\", font_style_name = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", background = \"[255,255,25 5]\"), Typesetting:-mo(\";\", form = \"infix\", fence = \"false\", sep arator = \"true\", lspace = \"0em\", rspace = \"thickmathspace\", stre tchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsi ze = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \+ \"false\", font_style_name = \"2D Input\", size = \"12\", foreground = \"[255,0,0]\", background = \"[255,255,255]\")), Typesetting:-mi(\"\" ));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6%-I#miGF$6 #Q!F'-F#6--F,6#Q%withF'-I#moGF$63Q\"(F'/%%formGQ'prefixF'/%&fenceGQ%tr ueF'/%*separatorGQ&falseF'/%'lspaceGQ.thinmathspaceF'/%'rspaceGFC/%)st retchyGF=/%*symmetricGF@/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(lar geopGF@/%.movablelimitsGF@/%'accentGF@/%0font_style_nameGQ)2D~InputF'/ %%sizeGQ#12F'/%+foregroundGQ*[255,0,0]F'/%+backgroundGQ.[255,255,255]F '-F,6#Q&plotsF'-F563Q\")F'/F9Q(postfixF'F;F>FA/FEQ2verythinmathspaceF' FFFHFJFMFPFRFTFVFYFfnFin-F563Q\":F'/F9Q&infixF'/F/FBQ/thickmathsp aceF'/FEF]p/FGF@FHFJFMFPFRFTFVFYFfnFin-I'mspaceGF$6&/%'heightGQ'0.0~ex F'/%&widthGQ'0.5~emF'/%&depthGFep/%*linebreakGQ%autoF'F1F4-F,6#Q(stude ntF'F_o-F563Q\";F'FioF[p/F?F=/FBQ$0emF'F^pF_pFHFJFMFPFRFTFVFYFfnFinF+" }}}{PARA 202 "" 0 "" {TEXT 264 11 "Exercise 1:" }}{PARA 202 "" 0 "" {TEXT 293 0 "" }}{PARA 202 "" 0 "" {TEXT 222 3 "(a)" }{TEXT 239 13 " \+ Plot f(" }{TEXT 282 1 "x" }{TEXT 239 1 "," }{TEXT 282 1 "y" }{TEXT 239 5 ") = x" }{TEXT 282 3 "y^3" }{TEXT 239 23 " sin(xy) for R = \+ \{(" }{TEXT 282 1 "x" }{TEXT 239 1 "," }{TEXT 282 1 "y" }{TEXT 239 5 " ) | " }{XPPEDIT 2 0 "Typesetting:-mrow(Typesetting:-mi(\"\"), Typeset ting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\" \"), Typesetting:-mverbatim(\"1\"\"!%\"xG\"), Typesetting:-mi(\"\")), \+ Typesetting:-mo(\"≤\", form = \"\", fence = \"false\", separator = \+ \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"false\", s ymmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits = \"false\", accent = \"false\", font_styl e_name = \"2D Math\", size = \"12\", foreground = \"[0,0,0]\", backgro und = \"[255,255,255]\"), Typesetting:-mverbatim(\"%#PiG\")), Typesett ing:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,TypesettingGI(_syslibGF '6%-I#miGF$6#Q!F'-F#6&F+-F#6%F+-I*mverbatimGF$6#Q)1\"\"!%\"xGF'F+-I#mo GF$63Q%≤F'/%%formGF./%&fenceGQ&falseF'/%*separatorGF?/%'lspaceGQ$0e mF'/%'rspaceGFD/%)stretchyGF?/%*symmetricGF?/%(maxsizeGQ)infinityF'/%( minsizeGQ\"1F'/%(largeopGF?/%.movablelimitsGF?/%'accentGF?/%0font_styl e_nameGQ(2D~MathF'/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+background GQ.[255,255,255]F'-F46#Q&%#PiGF'F+" }{TEXT 239 1 " " }{TEXT 239 1 " " }{TEXT 239 5 ", 0 " }{TEXT 211 1 "<" }{TEXT 239 1 " " }{TEXT 282 1 "y " }{TEXT 239 1 " " }{TEXT 211 1 "<" }{TEXT 239 7 " 1 \}. " }}{PARA 202 "" 0 "" {TEXT 222 4 "(b) " }{TEXT 239 27 " Approximate the integral \+ " }{XPPEDIT 239 0 "Typesetting:-mrow(Typesetting:-msubsup(Typesetting: -mo(\"∫\", form = \"prefix\", fence = \"false\", separator = \+ \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \"true\", sy mmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop \+ = \"true\", movablelimits = \"false\", accent = \"false\", font_style_ name = \"_cstyle25\", size = \"12\", foreground = \"[0,0,0]\", backgro und = \"[255,255,255]\"), Typesetting:-mi(\"a\"), Typesetting:-mi(\"b \"), superscriptshift = \"0\", subscriptshift = \"0\"), Typesetting:-m row(Typesetting:-msubsup(Typesetting:-mo(\"∫\", form = \"pref ix\", fence = \"false\", separator = \"false\", lspace = \"0em\", rspa ce = \"0em\", stretchy = \"true\", symmetric = \"false\", maxsize = \" infinity\", minsize = \"1\", largeop = \"true\", movablelimits = \"fal se\", accent = \"false\", font_style_name = \"_cstyle25\", size = \"12 \", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typeset ting:-mi(\"c\"), Typesetting:-mi(\"d\"), superscriptshift = \"0\", sub scriptshift = \"0\"), Typesetting:-mrow(Typesetting:-mi(\"f\"), Typese tting:-mo(\"⁡\", form = \"infix\", fence = \"false\", se parator = \"false\", lspace = \"0em\", rspace = \"0em\", stretchy = \" false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1 \", largeop = \"false\", movablelimits = \"false\", accent = \"false\" , font_style_name = \"_cstyle25\", size = \"12\", foreground = \"[0,0, 0]\", background = \"[255,255,255]\"), Typesetting:-mfenced(Typesettin g:-mrow(Typesetting:-mi(\"x\"), Typesetting:-mo(\",\", form = \"infix \", fence = \"false\", separator = \"true\", lspace = \"0em\", rspace \+ = \"verythickmathspace\", stretchy = \"false\", symmetric = \"false\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", movable limits = \"false\", accent = \"false\", font_style_name = \"_cstyle25 \", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,2 55]\"), Typesetting:-mi(\"y\")))), Typesetting:-mi(\"dA\"), Typesettin g:-mspace(height = \"0.0 ex\", width = \"0.5 em\", depth = \"0.0 ex\", linebreak = \"auto\"), Typesetting:-msemantics = \"int\"), Typesettin g:-mspace(height = \"0.0 ex\", width = \"0.5 em\", depth = \"0.0 ex\", linebreak = \"auto\"), Typesetting:-msemantics = \"int\");" "-I%mrowG 6#/I+modulenameG6\"I,TypesettingGI(_syslibGF'6&-I(msubsupGF$6'-I#moGF$ 63Q+∫F'/%%formGQ'prefixF'/%&fenceGQ&falseF'/%*separatorGF7/%' lspaceGQ$0emF'/%'rspaceGFQ3verythickmathspaceF'FfpFBFDFGFgpFLFNFPFSFVFY-Fg n6#Q\"yF'-Fgn6#Q#dAF'-I'mspaceGF$6&/%'heightGQ'0.0~exF'/%&widthGQ'0.5~ emF'/%&depthGFar/%*linebreakGQ%autoF'/I+msemanticsGF$Q$intF'F\\rFjr" } {TEXT 239 78 " with left and right Riemann double sums. Use m = 5 and n = 15 subdivisions." }}{PARA 202 "" 0 "" {TEXT 222 5 "(c) " }{TEXT 239 55 " Find the volume using iterated or double integral." }} {EXCHG {PARA 233 "> " 0 "" {TEXT 298 0 "" }}}{EXCHG {PARA 233 "> " 0 " " {TEXT 298 0 "" }}}{PARA 202 "" 0 "" {TEXT 264 11 "Exercise 2:" }} {PARA 202 "" 0 "" {TEXT 293 0 "" }}{PARA 202 "" 0 "" {TEXT 222 2 "a)" }{TEXT 239 7 " For " }{XPPEDIT 293 0 "Typesetting:-mrow(Typesetting: -mi(\"\"), Typesetting:-mrow(Typesetting:-mi(\"\"), Typesetting:-mverb atim(\"-%\"fG6$%\"xG%\"yG\"), Typesetting:-mo(\"=\", form = \"infix\", fence = \"false\", separator = \"false\", lspace = \"thickmathspace\" , rspace = \"thickmathspace\", stretchy = \"false\", symmetric = \"fal se\", maxsize = \"infinity\", minsize = \"1\", largeop = \"false\", mo vablelimits = \"false\", accent = \"false\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), Typesetting:-mrow(Typ esetting:-mi(\"\"), Typesetting:-msup(Typesetting:-mi(\"e\"), Typesett ing:-mrow(Typesetting:-mo(\"−\", form = \"infix\", fence = \"fal se\", separator = \"false\", lspace = \"mediummathspace\", rspace = \" mediummathspace\", stretchy = \"false\", symmetric = \"false\", maxsiz e = \"infinity\", minsize = \"1\", largeop = \"false\", movablelimits \+ = \"false\", accent = \"false\", font_style_name = \"_pstyle21\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255]\"), \+ Typesetting:-mfenced(Typesetting:-mrow(Typesetting:-mi(\"\"), Typesett ing:-msup(Typesetting:-mi(\"x\"), Typesetting:-mn(\"2\"), superscripts hift = \"0\"), Typesetting:-mo(\"+\", form = \"infix\", fence = \+ \"false\", separator = \"false\", lspace = \"mediummathspace\", rspace = \"mediummathspace\", stretchy = \"false\", symmetric = \"false\", m axsize = \"infinity\", minsize = \"1\", largeop = \"false\", movableli mits = \"false\", accent = \"false\", font_style_name = \"_pstyle21\", size = \"12\", foreground = \"[0,0,0]\", background = \"[255,255,255] \"), Typesetting:-msup(Typesetting:-mi(\"y\"), Typesetting:-mn(\"2\"), superscriptshift = \"0\"), Typesetting:-mi(\"\"))), Typesetting:-mi( \"\")), superscriptshift = \"0\"), Typesetting:-mi(\"\")), Typesetting :-mi(\"\")), Typesetting:-mi(\"\"));" "-I%mrowG6#/I+modulenameG6\"I,Ty pesettingGI(_syslibGF'6%-I#miGF$6#Q!F'-F#6'F+-I*mverbatimGF$6#Q0-%\"fG 6$%\"xG%\"yGF'-I#moGF$62Q\"=F'/%%formGQ&infixF'/%&fenceGQ&falseF'/%*se paratorGF>/%'lspaceGQ/thickmathspaceF'/%'rspaceGFC/%)stretchyGF>/%*sym metricGF>/%(maxsizeGQ)infinityF'/%(minsizeGQ\"1F'/%(largeopGF>/%.movab lelimitsGF>/%'accentGF>/%%sizeGQ#12F'/%+foregroundGQ([0,0,0]F'/%+backg roundGQ.[255,255,255]F'-F#6%F+-I%msupGF$6%-F,6#Q\"eF'-F#6%-F663Q(&minu s;F'F9F " 0 "" {TEXT 293 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {TEXT 293 0 "" }}}{PARA 202 "" 0 "" {TEXT 293 0 "" }}{PARA 202 "" 0 "" {TEXT 222 2 "b)" }{TEXT 239 26 " evaluate the \+ integral." }}{EXCHG {PARA 202 "> " 0 "" {TEXT 293 0 "" }}}{PARA 202 "" 0 "" {TEXT 239 2 "\n" }}{EXCHG {PARA 202 "> " 0 "" {TEXT 293 0 "" }}} {EXCHG {PARA 202 "> " 0 "" {TEXT 293 0 "" }}}}{SECT 1 {PARA 222 "" 0 " " {TEXT 216 15 "Another example" }{TEXT 216 2 " (" }{TEXT 217 16 "#58 \+ Section 15.3" }{TEXT 216 2 "):" }}{PARA 202 "" 0 "" {TEXT 293 0 "" }} {EXCHG {PARA 202 "" 0 "" {TEXT 239 37 "Graph the solid bounded by the \+ plane " }{TEXT 282 1 "x" }{TEXT 239 3 " + " }{TEXT 282 1 "y" }{TEXT 239 3 " + " }{TEXT 282 1 "z" }{TEXT 239 23 " =1 and the paraboloid " } {XPPEDIT 243 0 "z = 4-x^2-y^2;" "6#/I\"zG6\",(\"\"%\"\"\"*$)I\"xGF%\" \"#F(!\"\"*$)I\"yGF%F,F(F-" }{TEXT 239 1 " " }{TEXT 239 1 " " }{TEXT 239 28 " and find its exact volume." }}{PARA 202 "" 0 "" {TEXT 293 0 "" }}{PARA 202 "" 0 "" {TEXT 293 0 "" }}{PARA 202 "" 0 "" {TEXT 222 9 "Solution:" }}{PARA 202 "" 0 "" {TEXT 239 33 "We first graph the two f unctions:" }}{PARA 202 "" 0 "" {TEXT 293 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 13 "f:=4-x^2-y^2;" }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 209 15 "f := 4-x^2-y^2;" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 9 "g:=1-x-y;" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 51 "plot3d(\{f,g\},x=-2.5..2.5,y=-2.5..2.5,axes=boxed);" }}} {EXCHG {PARA 202 "" 0 "" {TEXT 239 35 "The domain of integration D is \+ the " }{TEXT 239 18 "projection on the " }{TEXT 282 2 "xy" }{TEXT 239 57 " plane of the curve of intersection of the two surfaces. " }} {PARA 202 "" 0 "" {TEXT 239 49 "To find this projection we need to eli minate the " }{TEXT 282 2 "z " }{TEXT 239 23 "from the two equations." }}{PARA 202 "" 0 "" {TEXT 239 12 "This gives " }{XPPEDIT 243 0 "1-x- y = 4-x^2-y^2;" "6#/,(\"\"\"F%I\"xG6\"!\"\"I\"yGF'F(,(\"\"%F%*$)F&\"\" #F%F(*$)F)F.F%F(" }{TEXT 239 1 " " }{TEXT 239 1 " " }{TEXT 239 6 " or " }{XPPEDIT 243 0 "x^2+y^2-x-y-3 = 0;" "6#/,,*$)I\"xG6\"\"\"#\"\"\"F **$)I\"yGF(F)F*F*F'!\"\"F-F.\"\"$F.\"\"!" }{TEXT 239 1 " " }{TEXT 239 1 " " }{TEXT 239 60 " . This is the equation of a circle (You can grap h it using " }{TEXT 239 89 "implicitplot to confirm that it is a circl e or you can complete the square which gives " }{XPPEDIT 243 0 "(x-1 /2)^2+(y-1/2)^2 = 7/2;" "6#/,&*$),&I\"xG6\"\"\"\"#F*\"\"#!\"\"F,F*F**$ ),&I\"yGF)F*F+F-F,F*F*#\"\"(F," }{TEXT 239 1 " " }{TEXT 239 1 " " } {TEXT 239 1 " " }{TEXT 239 2 ". " }}{PARA 202 "" 0 "" {TEXT 239 64 "In order to write the double integral we need to express either " }{TEXT 282 1 "x" }{TEXT 239 18 " as a function of " }{TEXT 282 2 "y " }{TEXT 239 3 "or " }{TEXT 282 1 "y" }{TEXT 239 18 " as a function of " } {TEXT 282 1 "x" }{TEXT 239 98 ". After completing the square this sou ld be fairly easy but we can have Maple do the work for us:" }}{PARA 202 "" 0 "" {TEXT 293 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 20 "funct:=solve(f=g,y);" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 27 "f1:=funct[2]; f2:=funct[1];" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 239 98 "Of course we get two functions: one for the lower bound \+ of the circle and one for the upper bound." }}{PARA 202 "" 0 "" {TEXT 239 25 "We also need to find the " }{TEXT 282 1 "x" }{TEXT 239 64 "- v alues corresponding to the intersections of these two curves " }}} {EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 23 "xlimit:=solve(f1=f2,x);" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 29 "x1:=xlimit[1]; x2:=x limit[2];" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 239 98 "To check that our findings are correct we plot the two surface together with the functi ons that we" }}{PARA 202 "" 0 "" {TEXT 239 120 "found as lower and upp er bound of the domain of integration. We also plot the curve of inter section of the two surfaces." }}{PARA 202 "" 0 "" {TEXT 239 117 "The p arametric equations of this curve of intersection were derived by para metrizing the circle and then substituting" }}{PARA 202 "" 0 "" {TEXT 239 20 "the expressions for " }{TEXT 282 1 "x" }{TEXT 239 5 " and " } {TEXT 282 1 "y" }{TEXT 239 50 " in the equation of the plane (or the p araboloid)." }}}{EXCHG {PARA 233 "> " 0 "" {MPLTEXT 1 209 16 "with(plo ttools):" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 10 "q:=plot3d(" }{MPLTEXT 1 227 7 "\{f,g\}" }{MPLTEXT 1 227 37 ",x=-2.5..2.5,y=-2.5.. 2.5,axes=boxed):" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 50 "p:= plot(\{f1,f2\},x=x1..x2,thickness=4,color=red):" }}}{EXCHG {PARA 233 " > " 0 "" {MPLTEXT 1 209 120 "r:=spacecurve([1/2+sqrt(7/2)*cos(t),1/2+s qrt(7/2)*sin(t),-sqrt(7/2)*(cos(t)+sin(t))],t=0..2*Pi,color=black,thic kness=2):" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 29 "h:=transfo rm((x,y)->[x,y,0]):" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 22 " display(\{q,r,h(p)\});" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 239 82 "From the graph above it looks like the red circle is indeed the projection on the " }{TEXT 282 2 "xy" }{TEXT 239 45 " plane of the intersection \+ of the two curves." }}{PARA 202 "" 0 "" {TEXT 239 77 "If we want anoth er check we can plot the two surfaces with domain the circle." }} {PARA 202 "" 0 "" {TEXT 293 0 "" }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 9 "plot3d(\{" }{MPLTEXT 1 227 40 "f,g\},x=x1 ..x2, y = \+ f1..f2,axes=boxed);" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 239 79 "We get \+ indeed the solid bounded above by the paraboloid and below by the pla ne" }}}{EXCHG {PARA 202 "" 0 "" {TEXT 239 44 "Now we can evaluate the \+ volume of the solid." }}}{EXCHG {PARA 202 "> " 0 "" {MPLTEXT 1 227 17 "V:=Doubleint(f-g," }{MPLTEXT 1 227 9 "y=f1..f2," }{MPLTEXT 1 227 1 "x " }{MPLTEXT 1 227 21 "=x1..x2): V=value(V);" }}}{EXCHG {PARA 233 "> " 0 "" {TEXT 298 0 "" }}}}{PARA 221 "" 0 "" {TEXT 299 0 "" }}{PARA 234 " " 0 "" {TEXT 300 0 "" }}{PARA 216 "" 0 "" {TEXT 301 0 "" }}} {MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }