{VERSION 5 0 "IBM INTEL NT" "5.0" } {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 "Hyperlink" -1 17 "" 0 1 0 128 128 1 2 0 1 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 "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 264 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{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" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "He ading 2" -1 4 1 {CSTYLE "" -1 -1 "Times" 1 14 0 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 "Heading 2" -1 256 1 {CSTYLE "" -1 -1 "Times" 1 14 0 0 0 1 2 1 2 2 2 2 1 1 1 1 }3 1 0 0 8 2 1 0 1 0 2 2 0 1 }{PSTYLE "Heading 2" -1 257 1 {CSTYLE "" -1 -1 "Time s" 1 14 0 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 "" 3 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {PARA 258 "" 0 "" {TEXT -1 16 "Triple Integrals" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "restart;\nwith(plots):" }}}{PARA 0 "" 0 "" {TEXT -1 261 "We visualize the three-dimensional domain of int egration of a triple integral, as well as how this domain is built as \+ the \"sum\" of a sequence of two-dimensional integrals.\nNote, however , that the animation is not an animation of the actual integration pro cess. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 72 "A few examples performing triple integration in Maple are also provid ed." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 38 "In tegrals are interpreted either with " }}{PARA 0 "" 0 "" {TEXT -1 9 "co nstant " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 11 "-limits: " } {XPPEDIT 18 0 "int(int(int(F(x,y,z),z = f[1](x,y) .. f[2](x,y)),y = g[ 1](x) .. g[2](x)),x = a .. b);" "6#-%$intG6$-F$6$-F$6$-%\"FG6%%\"xG%\" yG%\"zG/F/;-&%\"fG6#\"\"\"6$F-F.-&F46#\"\"#6$F-F./F.;-&%\"gG6#F66#F--& FA6#F;6#F-/F-;%\"aG%\"bG" }{TEXT -1 8 " or " }{XPPEDIT 18 0 "int(i nt(int(F(x,y,z),y = f[1](x,z) .. f[2](x,z)),z = g[1](x) .. g[2](x)),x \+ = a .. b);" "6#-%$intG6$-F$6$-F$6$-%\"FG6%%\"xG%\"yG%\"zG/F.;-&%\"fG6# \"\"\"6$F-F/-&F46#\"\"#6$F-F//F/;-&%\"gG6#F66#F--&FA6#F;6#F-/F-;%\"aG% \"bG" }{TEXT -1 4 " , " }}{PARA 0 "" 0 "" {TEXT -1 1 " " }}{PARA 0 " " 0 "" {TEXT -1 9 "constant " }{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 11 "-limits: " }{XPPEDIT 18 0 "int(int(int(F(x,y,z),z = f[1](x,y) .. f[2](x,y)),x = g[1](y) .. g[2](y)),y = a .. b);" "6#-%$intG6$-F$6$-F$ 6$-%\"FG6%%\"xG%\"yG%\"zG/F/;-&%\"fG6#\"\"\"6$F-F.-&F46#\"\"#6$F-F./F- ;-&%\"gG6#F66#F.-&FA6#F;6#F./F.;%\"aG%\"bG" }{TEXT -1 8 " or " } {XPPEDIT 18 0 "int(int(int(F(x,y,z),x = f[1](y,z) .. f[2](y,z)),z = g[ 1](y) .. g[2](y)),y = a .. b);" "6#-%$intG6$-F$6$-F$6$-%\"FG6%%\"xG%\" yG%\"zG/F-;-&%\"fG6#\"\"\"6$F.F/-&F46#\"\"#6$F.F//F/;-&%\"gG6#F66#F.-& FA6#F;6#F./F.;%\"aG%\"bG" }{TEXT -1 4 " , " }}{PARA 0 "" 0 "" {TEXT -1 9 "constant " }{XPPEDIT 18 0 "z" "6#%\"zG" }{TEXT -1 11 "-limits: \+ " }{XPPEDIT 18 0 "int(int(int(F(x,y,z),x = f[1](y,z) .. f[2](y,z)),y \+ = g[1](z) .. g[2](z)),z = a .. b);" "6#-%$intG6$-F$6$-F$6$-%\"FG6%%\"x G%\"yG%\"zG/F-;-&%\"fG6#\"\"\"6$F.F/-&F46#\"\"#6$F.F//F.;-&%\"gG6#F66# F/-&FA6#F;6#F//F/;%\"aG%\"bG" }{TEXT -1 8 " or " }{XPPEDIT 18 0 "i nt(int(int(F(x,y,z),y = f[1](x,z) .. f[2](x,z)),x = g[1](z) .. g[2](z) ),z = a .. b);" "6#-%$intG6$-F$6$-F$6$-%\"FG6%%\"xG%\"yG%\"zG/F.;-&%\" fG6#\"\"\"6$F-F/-&F46#\"\"#6$F-F//F-;-&%\"gG6#F66#F/-&FA6#F;6#F//F/;% \"aG%\"bG" }{TEXT -1 4 " . " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 153 "This worksheet is based mostly on work by Scot t Thatcher, who was a graduate student at Northwestern University at t he time I taught this material there." }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 139 "You can use this worksheet to help y ou with your visualization in general or to help with specific homewor k problems. For example, you can" }}{PARA 0 "" 0 "" {TEXT -1 134 " \+ (1) Check change-of-order-of-integration problems by comparing the \+ regions generated by the original limits and your new limits." }} {PARA 0 "" 0 "" {TEXT -1 84 " (2) Compare the region generated by your limits to a region given by the book." }}{PARA 0 "" 0 "" {TEXT -1 55 " (3) View an animation of the \"summation\" process." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "To see h ow to use the provided procedures, look at the examples provided in ea ch of the sections below." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 64 "Basic Procedure (to be executed before an ything below will work)" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 1769 "surf := proc () \n local x, y, z, i, j, s0, s1, t0, t1, s, t, \n pts, ind, params, mesh;\n\nmesh := 10:\nif (nargs > \+ 3) then \n mesh := args[4]:\n if (nargs > 4) then\n param s := seq (args [i], i=5..nargs)\n fi\nfi:\nind := seq (i/mesh, i=0. .mesh):\n\ns := op (1, args [2]):\nt := op (1, args [3]):\ns0 := unapp ly (op (1, op (2, args[2])), t):\ns1 := unapply (op (2, op (2, args[2] )), t):\nt0 := unapply (op (1, op (2, args[3])), s):\nt1 := unapply (o p (2, op (2, args[3])), s):\nx := unapply (op (1, args[1]), s, t):\ny \+ := unapply (op (2, args[1]), s, t):\nz := unapply (op (3, args[1]), s, t):\n\nif (type (s0(t), constant) and type (s1(t), constant))\n the n pts := [seq ([seq (\n [x ((1-i)*s0(t)+i*s1(t),\n \+ (1-j)*t0((1-i)*s0(t)+i*s1(t))+j*t1((1-i)*s0(t)+i*s1(t)) ),\n y ((1-i)*s0(t)+i*s1(t),\n (1-j )*t0((1-i)*s0(t)+i*s1(t))+j*t1((1-i)*s0(t)+i*s1(t))),\n \+ z ((1-i)*s0(t)+i*s1(t),\n (1-j)*t0((1-i)*s0(t)+i* s1(t))+j*t1((1-i)*s0(t)+i*s1(t)))],\n j=ind)],\n \+ i=ind)]\nelif (type (t0(s), constant) and type (t1(s), constant)) \n then pts := [seq ([seq (\n [x ((1-j)*s0((1-i)*t0(s) +i*t1(s))+j*s1((1-i)*t0(s)+i*t1(s)),\n (1-i)*t0(s)+ i*t1(s)),\n y ((1-j)*s0((1-i)*t0(s)+i*t1(s))+j*s1((1-i )*t0(s)+i*t1(s)),\n (1-i)*t0(s)+i*t1(s)),\n \+ z ((1-j)*s0((1-i)*t0(s)+i*t1(s))+j*s1((1-i)*t0(s)+i*t1(s)),\n (1-i)*t0(s)+i*t1(s))],\n j=ind)],\n \+ i=ind)]\nelse ERROR (`One parameter must have constant limits. `)\nfi;\npts := evalf (pts):\nif (nargs > 4)\n then RETURN (PLOT3D \+ (MESH(pts), params)) \n else RETURN (PLOT3D (MESH(pts)))\nfi:\nend: " }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT 256 22 "Integrals of the Form " }{XPPEDIT 18 0 "int (int (int (f(x,y,z), _), _), x)" "6#-%$intG6$-F$6$ -F$6$-%\"fG6%%\"xG%\"yG%\"zG%\"_GF0F-" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 257 20 "Definition of Limits" }}{PARA 0 "" 0 "" {TEXT -1 214 "The variable globres determines how many points are used to di splay each surface. Setting globres higher gives a better picture but will increase computation time. The default value of 10 is usually p retty good. " }}{PARA 0 "" 0 "" {TEXT -1 43 "Variables with subscript \+ zero, for example " }{XPPEDIT 18 0 "x[0]" "6#&%\"xG6#\"\"!" }{TEXT -1 66 ", are lower limits, and variables with subscript one, for example \+ " }{XPPEDIT 18 0 "x[1]" "6#&%\"xG6#\"\"\"" }{TEXT -1 21 ", are upper l imits. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "globres := 10:\nx0 := \+ (y, z) -> 0:\nx1 := (y, z) -> 1:\ny0 := (x, z) -> -x:\ny1 := (x, z) -> x:\nz0 := (x, y) -> -x:\nz1 := (x, y) -> x:" }}}{EXCHG {PARA 0 "" 0 " " {TEXT 258 20 "Display Solid Region" }}{PARA 0 "" 0 "" {TEXT -1 92 "W e take the limits defined above and display a solid picture of the reg ion of integration. " }}{PARA 0 "" 0 "" {TEXT -1 114 "The names top, \+ bot, north, south, east, and west are structures that represent the ou ter surfaces of the region. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 995 "t op := surf([x, y, z1(x,y)], x=x0(y, z1(x,y))..x1(y, z1(x,y)),\n \+ y=y0(x, z1(x,y))..y1(x, z1(x,y)),\n \+ globres):\nbot := surf([x, y, z0(x,y)], x=x0(y, z0(x,y ))..x1(y, z0(x,y)),\n y=y0(x, z0(x,y))..y1 (x, z0(x,y)),\n globres):\neast := surf([x , y1(x,z), z], x=x0(y1(x,z), z)..x1(y1(x,z), z),\n \+ z=z0(x, y1(x,z))..z1(x, y1(x,z)),\n \+ globres):\nwest := surf([x, y0(x,z), z], x=x0(y0(x,z), z)..x1(y0(x, z), z),\n z=z0(x, y0(x,z))..z1(x, y0(x,z)) ,\n globres):\nsouth := surf([x1(y,z), y, \+ z], y=y0(x1(y,z), z)..y1(x1(y,z), z),\n z= z0(x1(y,z), y)..z1(x1(y,z), y),\n globres) :\nnorth := surf([x0(y,z), y, z], y=y0(x0(y,z), z)..y1(x0(y,z), z),\n \+ z=z0(x0(y,z), y)..z1(x0(y,z), y),\n \+ globres):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 107 "To get a view of what the region looks like on the inside, try re moving one or two of these names from the " }{HYPERLNK 17 "display" 2 "display" "" }{TEXT -1 10 " command. " }}{PARA 0 "" 0 "" {TEXT -1 129 "Sometimes one of the sides is very thin, and removing it doesn't chan ge the picture. If this happens, try removing another side." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "display ([top, bot, north, south, east, w est], axes = framed, labels =[x,y,z], style=patch);" }{TEXT -1 0 "" }} }{EXCHG {PARA 0 "" 0 "" {TEXT 259 32 "Display Cross-Sections of Region " }}{PARA 0 "" 0 "" {TEXT -1 57 "Sometimes the solid region is still h ard to figure out. " }}{PARA 0 "" 0 "" {TEXT -1 96 "Here we give anot her viewing option by displaying cross-sections of the region of integ ration. " }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 247 "frame := display3d ([ top, bot, east, west], style=wireframe, color=black):\ncrsec := [seq ( display3d([surf ([i, y, z], y=y0(i, z)..y1(i,z), z=z0(i, y)..z1(i, y), globres, STYLE(PATCH)), frame]), i=seq((1-(n/10))*x0(y,z)+(n/10)*x1(y ,z), n=0..10))]: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 66 "If you repla ce \"insequence = false\" by \"insequence = true\" in the " } {HYPERLNK 17 "display" 2 "display" "" }{TEXT -1 206 " command in this \+ section, you will get an animation that shows graphically the integrat ion with respect to the outer variable. Each cross-section then repre sents one iteration of the inner double integral." }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 69 "display3d (crsec, insequence = false, axes=framed, \+ labels = [x,y,z]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 257 "" 0 "" {TEXT 260 22 "Integral s of the Form " }{XPPEDIT 18 0 "int (int (int (f(x,y,z), _), _), y)" " 6#-%$intG6$-F$6$-F$6$-%\"fG6%%\"xG%\"yG%\"zG%\"_GF0F." }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 261 20 "Definition of Limits" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 142 "globres := 10:\nx0 := (y, z) -> -y^2+z^2 :\nx1 := (y, z) -> y^2+z^2:\ny0 := (x, z) -> 0:\ny1 := (x, z) -> 1:\nz 0 := (x, y) -> -y:\nz1 := (x, y) -> y:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 262 20 "Display Solid Region" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 995 "top := surf([x, y, z1(x,y)], x=x0(y, z1(x,y))..x1(y, z1(x,y)),\n \+ y=y0(x, z1(x,y))..y1(x, z1(x,y)),\n \+ globres):\nbot := surf([x, y, z0(x,y)], x=x0(y, \+ z0(x,y))..x1(y, z0(x,y)),\n y=y0(x, z0(x,y ))..y1(x, z0(x,y)),\n globres):\neast := s urf([x, y1(x,z), z], x=x0(y1(x,z), z)..x1(y1(x,z), z),\n \+ z=z0(x, y1(x,z))..z1(x, y1(x,z)),\n \+ globres):\nwest := surf([x, y0(x,z), z], x=x0(y0(x,z), z)..x1 (y0(x,z), z),\n z=z0(x, y0(x,z))..z1(x, y0 (x,z)),\n globres):\nsouth := surf([x1(y,z ), y, z], y=y0(x1(y,z), z)..y1(x1(y,z), z),\n \+ z=z0(x1(y,z), y)..z1(x1(y,z), y),\n gl obres):\nnorth := surf([x0(y,z), y, z], y=y0(x0(y,z), z)..y1(x0(y,z), \+ z),\n z=z0(x0(y,z), y)..z1(x0(y,z), y),\n \+ globres):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "display ([top, bot, north, south, east, west], axes = framed, labels =[x,y,z], style=patch);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 263 32 "Display Cross-Sections of Region" } {TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 248 "frame := display3d ([top, bot, north, south], style=wireframe, color=black):\ncrsec := [ seq (display3d([surf ([x, i, z], x=x0(i, z)..x1(i,z), z=z0(x, i)..z1(x , i), globres, STYLE(PATCH)), frame]), i=seq((1-(n/10))*y0(y,z)+(n/10) *y1(y,z), n=0..10))]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "di splay3d (crsec, insequence = false, axes=framed, labels = [x,y,z]);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 264 22 "Integrals of the Form " }{XPPEDIT 18 0 "int (int (int (f(x,y,z), _), _), z)" "6#-%$intG6$-F$6$-F$6$-%\"fG6% %\"xG%\"yG%\"zG%\"_GF0F/" }{TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT 265 20 "Definition of Limits" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 183 "globres := 10:\nx0 := (y, z) -> -sqrt (1-y^2-z^2):\nx1 := (y, z) \+ -> sqrt (1-y^2-z^2):\ny0 := (x, z) -> -sqrt (1-z^2):\ny1 := (x, z) -> \+ sqrt (1-z^2):\nz0 := (x, y) -> -1:\nz1 := (x, y) -> 1:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 266 20 "Display Solid Region" }}{PARA 0 "> " 0 " " {MPLTEXT 1 0 995 "top := surf([x, y, z1(x,y)], x=x0(y, z1(x,y))..x1( y, z1(x,y)),\n y=y0(x, z1(x,y))..y1(x, z1( x,y)),\n globres):\nbot := surf([x, y, z0( x,y)], x=x0(y, z0(x,y))..x1(y, z0(x,y)),\n \+ y=y0(x, z0(x,y))..y1(x, z0(x,y)),\n globr es):\neast := surf([x, y1(x,z), z], x=x0(y1(x,z), z)..x1(y1(x,z), z), \n z=z0(x, y1(x,z))..z1(x, y1(x,z)),\n \+ globres):\nwest := surf([x, y0(x,z), z], x=x0 (y0(x,z), z)..x1(y0(x,z), z),\n z=z0(x, y0 (x,z))..z1(x, y0(x,z)),\n globres):\nsouth := surf([x1(y,z), y, z], y=y0(x1(y,z), z)..y1(x1(y,z), z),\n \+ z=z0(x1(y,z), y)..z1(x1(y,z), y),\n \+ globres):\nnorth := surf([x0(y,z), y, z], y=y0(x0(y,z), \+ z)..y1(x0(y,z), z),\n z=z0(x0(y,z), y)..z1 (x0(y,z), y),\n globres):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 92 "display ([top, bot, north, south, e ast, west], axes = framed, labels =[x,y,z], style=patch);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 267 32 "Display Cross-Sections of Region" }{TEXT -1 0 "" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 606 "frame := display3d ([north, south, east, west], style=wireframe, color=black): \nif (type (y0(x,z), constant) and type (y1 (x,z), constant))\n then \+ crsec := [seq (display3d([surf ([x, i, z], x=x0(i, z)..x1(i,z),\n \+ z=z0(x, i)..z1(x, i), globres, STYLE(PATCH)), frame]),\n \+ i=seq((1-(n/10))*y0(y,z)+(n/10)*y1(y,z), n=0..10))]\nelif (type (z0 (x,y), constant) and type (z1 (x,y), constant))\n then crsec := [seq \+ (display3d([surf ([x, y, i], x=x0(i, y)..x1(i,y),\n y=y0(x, \+ i)..y1(x, i), globres, STYLE(PATCH)), frame]),\n i=seq((1-(n /10))*z0(x,z)+(n/10)*z1(x,z), n=0..10))] \nfi: " }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 69 "display3d (crsec, insequence = false, axes=fra med, labels = [x,y,z]);" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 40 "How to Comp ute Triple Integrals in Maple" }}{PARA 0 "" 0 "" {TEXT -1 93 "If you s imply want to evaluate a triple integral, you can do this by nesting t he int command." }}{PARA 0 "" 0 "" {TEXT -1 72 "For the previous three domains, with the function F(x,y,z) = x*y this is" }}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 24 "F := (x,y,z) -> x^2*z^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 102 "Int(Int(Int(F(x,y,z), y=-x..x), z=-x..x), \+ x=0..1) = int(int(int(F(x,y,z), y=-x..x), z=-x..x), x=0..1);" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 126 "Int(Int(Int(F(x,y,z), x=-y^ 2+z^2..y^2+z^2), z=-y..y), y=0..1) = int(int(int(F(x,y,z), x=-y^2+z^2. .y^2+z^2), z=-y..y), y=0..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 200 "Int(Int(Int(F(x,y,z), x=-sqrt(1-y^2-z^2)..sqrt(1-y^2-z^2)), y=- sqrt(1-z^2)..sqrt(1-z^2)), z=-1..1) = int(int(int(F(x,y,z), x=-sqrt(1- y^2-z^2)..sqrt(1-y^2-z^2)), y=-sqrt(1-z^2)..sqrt(1-z^2)), z=-1..1);" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{MARK "21" 0 } {VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }