{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 "" 1 10 0 0 0 0 0 0 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 1 0 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 1 0 0 0 0 0 0 0 0 1 } {CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 267 "Helvetica" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 272 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 273 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 1 }{CSTYLE "" -1 275 "" 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 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 3" -1 5 1 {CSTYLE "" -1 -1 "Times " 1 12 0 0 0 1 1 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" -1 256 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 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 0 "" }{TEXT 271 0 "" }{TEXT 272 23 "Calculus IV with Maple\n" }{TEXT 273 32 "Copyright 2002, Dr. J ack Wagner\n" }{TEXT -1 30 "j.wagner@intelligentsearch.com" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 22 "Lesson 9: Surface Area" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 9 "Topics: T" }{TEXT 256 196 "he surface area is ca lculated five different ways including the use of the first fundament al form, the image of the coordinate plane under the map defining the surface and surfaces of rotation. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 262 "We compute the area of a surfa ce by dividing it up into small parallelograms, and taking the limit o f the sum of their areas as the parallelograms are made progressively \+ smaller. One way to look at this is to recognize that the area of a p arallelogram of sides " }{TEXT 265 1 "a" }{TEXT -1 5 " and " }{TEXT 266 1 "b" }{TEXT -1 21 " with included angle " }{XPPEDIT 18 0 "gamma; " "6#%&gammaG" }{TEXT -1 5 ", is " }{XPPEDIT 18 0 "absin(gamma);" "6#- %&absinG6#%&gammaG" }{TEXT -1 43 ". Consider a surface known by a fun ction, " }{XPPEDIT 18 0 "phi(u,v);" "6#-%$phiG6$%\"uG%\"vG" }{TEXT -1 54 " . The area of a small rectangle in the u-v plane is " }{XPPEDIT 18 0 "Delta*u*Delta*v;" "6#**%&DeltaG\"\"\"%\"uGF%F$F%%\"vGF%" }{TEXT -1 62 " . The area of the image of this rectangle on the surface is \+ " }}{PARA 0 "" 0 "" {XPPEDIT 18 0 "phi[u](u,v)*Delta*u*phi[v](u,v)*Del ta*v;" "6#*.-&%$phiG6#%\"uG6$F(%\"vG\"\"\"%&DeltaGF+F(F+-&F&6#F*6$F(F* F+F,F+F*F+" }{XPPEDIT 18 0 "sin(gamma[p]);" "6#-%$sinG6#&%&gammaG6#%\" pG" }{TEXT -1 9 " , where " }{XPPEDIT 18 0 "phi[u](u,v) = diff(phi(u,v ),u);" "6#/-&%$phiG6#%\"uG6$F(%\"vG-%%diffG6$-F&6$F(F*F(" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "gamma[p];" "6#&%&gammaG6#%\"pG" }{TEXT -1 18 " is the image of " }{XPPEDIT 18 0 "gamma;" "6#%&gammaG" }{TEXT -1 8 ". Then, " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }{XPPEDIT 18 0 "phi[u](u,v) *Delta*u*phi[v](u,v)*Delta*v;" "6#*.-&%$phiG6#%\"uG6$F(%\"vG\"\"\"%&De ltaGF+F(F+-&F&6#F*6$F(F*F+F,F+F*F+" }{XPPEDIT 18 0 "sin(gamma[p]);" "6 #-%$sinG6#&%&gammaG6#%\"pG" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "phi[u](u ,v)*phi[v](u,v);" "6#*&-&%$phiG6#%\"uG6$F(%\"vG\"\"\"-&F&6#F*6$F(F*F+ " }{XPPEDIT 18 0 "sin(gamma[p]);" "6#-%$sinG6#&%&gammaG6#%\"pG" } {TEXT -1 0 "" }{XPPEDIT 18 0 "Delta*u" "6#*&%&DeltaG\"\"\"%\"uGF%" } {XPPEDIT 18 0 "Delta*v;" "6#*&%&DeltaG\"\"\"%\"vGF%" }{TEXT -1 5 " = | " }{XPPEDIT 18 0 "phi[u](u,v);" "6#-&%$phiG6#%\"uG6$F'%\"vG" }{TEXT -1 0 "" }{TEXT 267 1 "x" }{XPPEDIT 18 0 "phi[v](u,v);" "6#-&%$phiG6#% \"vG6$%\"uGF'" }{TEXT -1 1 "|" }{XPPEDIT 18 0 "Delta*u*Delta*v;" "6#** %&DeltaG\"\"\"%\"uGF%F$F%%\"vGF%" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "and the limit of the Riem ann sum will be " }{XPPEDIT 18 0 "int(int(abs(phi[u](u,v)*x*phi[v](u, v)),u = u[0] .. u[1]),v = v[0] .. v[1]);" "6#-%$intG6$-F$6$-%$absG6#*( -&%$phiG6#%\"uG6$F0%\"vG\"\"\"%\"xGF3-&F.6#F26$F0F2F3/F0;&F06#\"\"!&F0 6#F3/F2;&F26#F=&F26#F3" }{TEXT -1 66 " . \+ " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 257 11 "Example 9.1" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "Find the area o f the surface defined by: " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "p hi(u,v) = u^2-v^3;" "6#/-%$phiG6$%\"uG%\"vG,&*$F'\"\"#\"\"\"*$F(\"\"$! \"\"" }{TEXT -1 5 ", \{" }{XPPEDIT 18 0 "-1 <= u;" "6#1,$\"\"\"!\"\" %\"uG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "v <= 1;" "6#1%\"vG\"\"\"" } {TEXT -1 1 "\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "restart: \+ with(plots): with(LinearAlgebra): with(VectorCalculus):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f := ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "plot3d(f, u=-1..1, v=-1..1, axes=fr amed, color=blue, style=wireframe);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 148 "Maple's plotting function makes very concrete the idea o f a surface composed of parallelograms that are the images of rectangl es in the u-v plane.\011\011" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "df[u] := diff(f, u);\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "df[v] := diff(f, v);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "areaElementVector := df[u] &x df[v];" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "dA := Norm( areaElementVector, 2 );" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "dA := simplify( dA ) assuming real; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "int(int(dA, u=-1..1), v =-1..1);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 121 "If you attempt to explicitly integrate this expression, \+ both you and Maple will fail. Numerical evaluation is necessary." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(%);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 258 11 "Example \+ 9.2" }}{PARA 0 "" 0 "" {TEXT 274 3 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Find the surface area of " }{XPPEDIT 18 0 "phi(u,v) = sin (u)-cos(v);" "6#/-%$phiG6$%\"uG%\"vG,&-%$sinG6#F'\"\"\"-%$cosG6#F(!\" \"" }{TEXT -1 4 " , \{" }{XPPEDIT 18 0 "-1 <= u;" "6#1,$\"\"\"!\"\"%\" uG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "v <= 1;" "6#1%\"vG\"\"\"" }{TEXT -1 1 "\}" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "restart: with(p lots): with(LinearAlgebra): with(VectorCalculus):" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 28 "f := ;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "plot3d(f, u=-1..1, v=-1..1, axes=fr amed, color=blue, style=wireframe, labels=[u, v, z]);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "Now consider two vectors, " }{XPPEDIT 18 0 "v[1];" "6#&%\"vG6#\"\"\"" }{TEXT -1 6 " and " }{XPPEDIT 18 0 "v[2] ;" "6#&%\"vG6#\"\"#" }{TEXT -1 99 " that form two sides of a parallel ogram on a surface. Form a matrix with the vectors as columns. " } {XPPEDIT 18 0 "D = matrix([[v[1], v[2]], [%?, %?]]);" "6#/%\"DG-%'matr ixG6#7$7$&%\"vG6#\"\"\"&F+6#\"\"#7$%#%?GF2" }{TEXT -1 23 " Take the product, " }{XPPEDIT 18 0 "D^T;" "6#)%\"DG%\"TG" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "matrix([[v[1], %?], [v[2], %?]]);" "6#-%'matrixG6#7$7$& %\"vG6#\"\"\"%#%?G7$&F)6#\"\"#F," }{TEXT -1 1 " " }{XPPEDIT 18 0 "matr ix([[v[1], v[2]], [%?, %?]]);" "6#-%'matrixG6#7$7$&%\"vG6#\"\"\"&F)6# \"\"#7$%#%?GF0" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "matrix([[abs(v[1])^2 , v[1]*v[2]], [v[1]*v[2], abs(v[2])^2]]);" "6#-%'matrixG6#7$7$*$-%$abs G6#&%\"vG6#\"\"\"\"\"#*&&F-6#F/F/&F-6#F0F/7$*&&F-6#F/F/&F-6#F0F/*$-F*6 #&F-6#F0F0" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 41 "Compute the determinant of this product. " }{XPPEDIT 18 0 "Det(D^T*D);" "6#-%$Det G6#*&)%\"DG%\"TG\"\"\"F(F*" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "abs(v[1] )^2*abs(v[2])^2-(v[1]*v[2])^2;" "6#,&*&-%$absG6#&%\"vG6#\"\"\"\"\"#-F& 6#&F)6#F,F,F+*$*&&F)6#F+F+&F)6#F,F+F,!\"\"" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "abs(v[1])^2*abs(v[2])^2-abs(v[1])^2*abs(v[2])^2*cos(v[1 ],v[2])^2;" "6#,&*&-%$absG6#&%\"vG6#\"\"\"\"\"#-F&6#&F)6#F,F,F+*(-F&6# &F)6#F+F,-F&6#&F)6#F,F,-%$cosG6$&F)6#F+&F)6#F,F,!\"\"" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 2 "= " }{XPPEDIT 18 0 "abs(v[1])^2*abs(v[2] )^2*(1-cos(v[1],v[2])^2);" "6#*(-%$absG6#&%\"vG6#\"\"\"\"\"#-F%6#&F(6# F+F+,&F*F**$-%$cosG6$&F(6#F*&F(6#F+F+!\"\"F*" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "abs(v[1])^2*abs(v[2])^2*sin(v[1],v[2])^2;" "6#*(-%$absG 6#&%\"vG6#\"\"\"\"\"#-F%6#&F(6#F+F+-%$sinG6$&F(6#F*&F(6#F+F+" }{TEXT -1 4 " = " }{XPPEDIT 18 0 "abs(v[1]*x*v[2])^2;" "6#*$-%$absG6#*(&%\"v G6#\"\"\"F+%\"xGF+&F)6#\"\"#F+F/" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "dA ^2;" "6#*$%#dAG\"\"#" }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 9 "Le tting " }{XPPEDIT 18 0 "v[1] = f[v];" "6#/&%\"vG6#\"\"\"&%\"fG6#F%" } {TEXT -1 5 " and " }{XPPEDIT 18 0 "v[2] = f[u];" "6#/&%\"vG6#\"\"#&%\" fG6#%\"uG" }{TEXT -1 48 ", this yields another formula for surface are a. " }{XPPEDIT 18 0 "A = Int(Int(sqrt(det(J[f]^T*J[f])),u),v);" "6#/% \"AG-%$IntG6$-F&6$-%%sqrtG6#-%$detG6#*&)&%\"JG6#%\"fG%\"TG\"\"\"&F36#F 5F7%\"uG%\"vG" }{TEXT -1 7 " where " }{XPPEDIT 18 0 "J[f];" "6#&%\"JG6 #%\"fG" }{TEXT -1 28 " is the Jacobian matrix of " }{TEXT 268 1 "f" } {TEXT -1 2 ". " }}}{EXCHG {PARA 0 "" 0 "" {HYPERLNK 17 "Jacobian" 2 "J acobian" "" }{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "J := Jacobian(f, [u, v, w]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "J := DeleteColumn(J, [3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "S := Transpose(J).J;\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 15 "Determinant(S);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Int(Int(sqrt(Determinant(S)), u=-1..1), v=-1..1);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "evalf(%);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 77 "Now recol lect that the principal parts of the tangent vectors to a surface, " }{XPPEDIT 18 0 "F(u,v);" "6#-%\"FG6$%\"uG%\"vG" }{TEXT -1 36 ", in the coordinate directions, are " }{XPPEDIT 18 0 "F[u];" "6#&%\"FG6#%\"uG " }{TEXT -1 6 " and " }{XPPEDIT 18 0 "F[v];" "6#&%\"FG6#%\"vG" } {TEXT -1 35 ". Also recollect the definitions; " }{XPPEDIT 18 0 "E = \+ anglebracket(F[u],F[u]);" "6#/%\"EG-%-anglebracketG6$&%\"FG6#%\"uG&F)6 #F+" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "F = anglebracket(F[u],F[v]);" "6 #/%\"FG-%-anglebracketG6$&F$6#%\"uG&F$6#%\"vG" }{TEXT -1 2 ", " } {XPPEDIT 18 0 "G = anglebracket(F[v],F[v]);" "6#/%\"GG-%-anglebracketG 6$&%\"FG6#%\"vG&F)6#F+" }{TEXT -1 12 ". Then, from" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }{XPPEDIT 18 0 "abs(v[1])^2*abs(v[2])^2-(v[1]*v[2])^2; " "6#,&*&-%$absG6#&%\"vG6#\"\"\"\"\"#-F&6#&F)6#F,F,F+*$*&&F)6#F+F+&F)6 #F,F+F,!\"\"" }{TEXT -1 5 " = " }{XPPEDIT 18 0 "abs(v[1]*x*v[2])^2; " "6#*$-%$absG6#*(&%\"vG6#\"\"\"F+%\"xGF+&F)6#\"\"#F+F/" }{TEXT -1 51 " , we obtain yet another formula for surface area, " }}{PARA 0 "" 0 " " {XPPEDIT 18 0 "Int(Int(sqrt(E*G-F^2),u),v);" "6#-%$IntG6$-F$6$-%%sqr tG6#,&*&%\"EG\"\"\"%\"GGF.F.*$%\"FG\"\"#!\"\"%\"uG%\"vG" }{TEXT -1 80 " This is an important relation because the area is expressed in term s that are " }{TEXT 269 9 "intrinsic" }{TEXT -1 16 " to the surface." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 259 11 "Example 9.3" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "Find the surface area defined by: " }{XPPEDIT 18 0 "f(u,v) = [u^2+v^2, sin(u), cos(v)];" "6#/-%\"fG6$%\"uG%\"vG7%,&*$F' \"\"#\"\"\"*$F(F,F--%$sinG6#F'-%$cosG6#F(" }{TEXT -1 3 " , " } {XPPEDIT 18 0 "0 <= u;" "6#1\"\"!%\"uG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "v <= Pi/2;" "6#1%\"vG*&%#PiG\"\"\"\"\"#!\"\"" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 64 "restart: with(plots): with(LinearAlgebra): wit h(VectorCalculus):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "f := \+ ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "plot3d(f, u=0..Pi/2, v=0..Pi/2, axes=framed, color=blue, style=wir eframe, labels=[u, v, z]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "df[u] := diff(f, u);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "df[v] := diff(f, v);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "E := df[u].df[u];\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "F := df[v].df[u];\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 " G := df[v].df[v];\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "Int( Int(sqrt(E*G-F^2), u=0..Pi/2), v=0..Pi/2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "evalf(%); \n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 11 "Example 9.4" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 99 "Consider the su rface of revolution resulting from revolving the curve whose parametri c equation is " }{XPPEDIT 18 0 "f(t) = [x(t), y(t)];" "6#/-%\"fG6#%\"t G7$-%\"xG6#F'-%\"yG6#F'" }{TEXT -1 22 " , around the x axis. " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 9 " " }{XPPEDIT 18 0 "F(t,thet a) = [x(t), y(t)*cos(theta), y(t)*sin(theta)];" "6#/-%\"FG6$%\"tG%&the taG7%-%\"xG6#F'*&-%\"yG6#F'\"\"\"-%$cosG6#F(F1*&-F/6#F'F1-%$sinG6#F(F1 " }{TEXT -1 1 " " }}{PARA 0 "" 0 "" {TEXT -1 39 "We wish to calculate \+ its surface area. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "resta rt: with(plots): with(LinearAlgebra): with(VectorCalculus):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "f := ;\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "F := ;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "dF[ t] := diff(F, t);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 "dF[t heta] := diff(F, theta); \n\n\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "XP := dF[t] &x dF[theta];\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "Nsq := simplify(XP.XP, trig);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "dA := sqrt(factor(Nsq), symbolic);\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 53 "A \+ very pretty result. Continuing in the same spirit;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "df[t] := diff(f, t);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "ds := Norm(df[t], 2);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "From which we c onclude that " }{XPPEDIT 18 0 "dA = y(t)*ds;" "6#/%#dAG*&-%\"yG6#%\"tG \"\"\"%#dsGF*" }{TEXT -1 36 " . Therefore, we find the area as, " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "Area := Int(Int(y(t)*ds, t=a ..b), theta=0..2*Pi);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 " value(%);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "A result familiar \+ from the calculus of a single variable." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 12 "Example 9. 5" }}{PARA 0 "" 0 "" {TEXT 275 1 " " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 62 "Find th e area of the surface formed by rotating the graph of " }{XPPEDIT 18 0 "f(t) = [sqrt(t)/(1+t), sin(t)];" "6#/-%\"fG6#%\"tG7$*&-%%sqrtG6#F' \"\"\",&F-F-F'F-!\"\"-%$sinG6#F'" }{TEXT -1 4 " , " }{XPPEDIT 18 0 "t = 0 .. 10;" "6#/%\"tG;\"\"!\"#5" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "restart: with(plots): with(LinearAlgebra): with(Vecto rCalculus):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "f := ;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot ([f[1], f[2], t=0..10]);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 408 "He re we see one of Maple's advantages. By plotting the curve before we \+ start, we immediately know that there is a problem of self intersectio n when t>1.0. To find the area of such a solid of rotation we need to break it up into simple (non self intersecting) surfaces. We will se parately compute the area of the surface formed by rotating the curve \+ around the x axis from t=0 to t=1 and from t=1 to t=10. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "plot([f[1], f[2], t=0..1]);\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "plot([f[1], f[2], t=1..10]); \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "F := [f[1], f[2]*cos( theta), f[2]*sin(theta)];\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 82 "plot3d(F, t=0..10, theta=0..2*Pi, axes=normal, grid=[50,50], orien tation=[80,60]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "df[t] : = diff(f, t);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "ds[t] := Norm(df[t], 2); \n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "2*Pi *Int(f[2]*ds[t], t=0..1);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "A[1] := evalf(%);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 " A[2] := evalf(2*Pi*Int(f[2]*ds[t], t=1..10));\n" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 19 "Area := A[1]+A[2];\n" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 30 " From the equation, " }{TEXT 262 5 "Area " } {TEXT -1 2 "= " }{XPPEDIT 18 0 "int(int(abs(phi[u](u,v)*x*phi[v](u,v)) ,u = u[0] .. u[1]),v = v[0] .. v[1])" "6#-%$intG6$-F$6$-%$absG6#*(-&%$ phiG6#%\"uG6$F0%\"vG\"\"\"%\"xGF3-&F.6#F26$F0F2F3/F0;&F06#\"\"!&F06#F3 /F2;&F26#F=&F26#F3" }{TEXT -1 101 " we may derive yet another express ion for area. Let a surface be known by an equation of the form, " } {XPPEDIT 18 0 "z = f(x,y);" "6#/%\"zG-%\"fG6$%\"xG%\"yG" }{TEXT -1 4 " or " }{XPPEDIT 18 0 "phi(x,y) = [x, y, f(x,y)];" "6#/-%$phiG6$%\"xG% \"yG7%F'F(-%\"fG6$F'F(" }{TEXT -1 2 " " }}}{EXCHG {PARA 0 "" 0 "" {XPPEDIT 18 0 "A = Int(Int(phi[x]*x*phi[y],x),y);" "6#/%\"AG-%$IntG6$- F&6$*(&%$phiG6#%\"xG\"\"\"F.F/&F,6#%\"yGF/F.F2" }{TEXT -1 3 " = " } {XPPEDIT 18 0 "Int(Int(abs([1, 0, f[x]]*x*[0, 1, f[y]]),x),y);" "6#-%$ IntG6$-F$6$-%$absG6#*(7%\"\"\"\"\"!&%\"fG6#%\"xGF-F2F-7%F.F-&F06#%\"yG F-F2F6" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(Int(sqrt(1+f[x]^2+f[y]^2 ),x),y);" "6#-%$IntG6$-F$6$-%%sqrtG6#,(\"\"\"F,*$&%\"fG6#%\"xG\"\"#F,* $&F/6#%\"yGF2F,F1F6" }{TEXT -1 3 " = " }{XPPEDIT 18 0 "Int(Int(sqrt(1+ abs(grad(f))^2),x),y);" "6#-%$IntG6$-F$6$-%%sqrtG6#,&\"\"\"F,*$-%$absG 6#-%%gradG6#%\"fG\"\"#F,%\"xG%\"yG" }{TEXT -1 0 "" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 263 12 "Example 9. 6" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "restart: with(plots):with(LinearAlgebra): with(VectorCalculus): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Z := log(x+y);\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 125 "p1 := implicitplot3d(\{x=.5 , x=1.5, y=.5, y=1.5\}, x=.5..1.5, y=.5..1.5, z=\n-1..1.5, axes=framed , color=red, style=wireframe):\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 75 "p2 := plot3d(Z, x=0..2, y=0..2, axes=framed, labels=[ x, y, z], color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "d isplay(p1, p2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "gZ := \+ Gradient(Z, [x, y]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "d A := sqrt(1+gZ.gZ);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "ev alf(Int(Int(dA, y=.5..1.5), x=0.5..1.5));\n" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 409 "We now have five expressions for surface area.\011Unfo rtunately, most surfaces yield integrals which are difficult or imposs ible to evaluate in closed or series form. Even numerical evaluation m ay be extraordinarily difficult. However, the use of the definition of the definite integral may succeed when all else fails. We construct u pper and lower Riemann sums and examine a sequence of values for conve rgence. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 264 11 "Example 9.7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "Find the area of the surface described by f(x, y)=[sin(x), sin(y), sin(x+y)]. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "restart: with(plots): with(LinearAlgebra): with(Vecto rCalculus):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "f := ;\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "p lot3d(f, x=-Pi..Pi, y=-Pi..Pi, axes=framed, grid=[70,70], style=patchn ogrid, lightmodel=light2);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "J := Jacobian(f, [x, y, z]);\n" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "J := DeleteColumn(J, [3]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "dA := sqrt(Determinant(Transpose(J).J));\n" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "Int(Int(dA, x=0..2*Pi), y=0. .2*Pi);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 127 "Maple has difficult y integrating this expression or even producing a numerical answer. \+ \nFirst, let's see what dA looks like.\n" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 65 "plot3d(dA, x=0..2*Pi, y=0..2*Pi, axes=framed, color =aquamarine);\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 74 "Noting the sym metry of the problem we need only evaluate the integral for " } {XPPEDIT 18 0 "0 <= x;" "6#1\"\"!%\"xG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "y <= Pi;" "6#1%\"yG%#PiG" }{TEXT -1 5 ". " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 56 "We create Reimann sums using intervals over x and y \+ of " }{XPPEDIT 18 0 "Pi/(10*s);" "6#*&%#PiG\"\"\"*&\"#5F%%\"sGF%!\"\" " }{TEXT -1 103 " where s increases from 1 to 8. This gives a progres sion from 100 to 6400 cells, each with an area of " }{XPPEDIT 18 0 "Pi ^2/((10*s)^2);" "6#*&%#PiG\"\"#*$*&\"#5\"\"\"%\"sGF)F%!\"\"" }{TEXT -1 68 " over which to evaluate and sum dA. First the lower Reimann s ums. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 180 "for s from 1 to 8 do \n F := (k, t)->subs(\{x=k*Pi/(10*s), y=t*Pi/(10*s)\}, dA):\n A| |s := evalf((Pi/(10*s))^2*sum(sum(F(p, q), p=0..10*s-1), q=0..10*s-1)) :\n print(A||s); \nend do:\n\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 " Now the upper Reimann sums." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 175 "for s from 1 to 8 do \n F := (k, t)->subs(\{x=k*P i/(10*s), y=t*Pi/(10*s)\}, dA): \n A||s := evalf((Pi/(10*s))^2*sum(su m(F(p, q), p=1..10*s), q=1..10*s)):\n print(A||s) \nend do:\n" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 54 "Convergence of both sequences to 7 .43 seems apparent. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "Are a := 4*7.43;\n" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 270 9 "Practice\n" } {TEXT -1 42 "\n1. Find the area of the plane defined by " }{XPPEDIT 18 0 "2*x-3*y+4*z = 10;" "6#/,(*&\"\"#\"\"\"%\"xGF'F'*&\"\"$F'%\"yGF'! \"\"*&\"\"%F'%\"zGF'F'\"#5" }{TEXT -1 46 " enclosed in the elliptic cy linder over, \n . " }{XPPEDIT 18 0 "y^2/4+x^2/5 = 20;" "6#/,&*&%\"yG \"\"#\"\"%!\"\"\"\"\"*&%\"xGF'\"\"&F)F*\"#?" }{TEXT -1 45 "\n\n2. Find the area of the surface defined by " }{XPPEDIT 18 0 "z = exp(x+y);" " 6#/%\"zG-%$expG6#,&%\"xG\"\"\"%\"yGF*" }{TEXT -1 45 " , that lies over the lozenge\n defined by: \n" }{XPPEDIT 18 0 "\{y = x+1/2, y = x-1/2 , y = -x+1/2, y = -x-1/2\};" "6#<&/%\"yG,&%\"xG\"\"\"*&F(F(\"\"#!\"\"F (/F%,&F'F(*&F(F(F*F+F+/F%,&F'F+*&F(F(F*F+F(/F%,&F'F+*&F(F(F*F+F+" } {TEXT -1 103 "\n3. Find the area of the surface obtained by rotating, \+ around the x axis, the curve \nparametrized by: " }{XPPEDIT 18 0 "f(t ) = [t/sqrt(1+t^2), t^2/sqrt(1+t)];" "6#/-%\"fG6#%\"tG7$*&F'\"\"\"-%%s qrtG6#,&F*F**$F'\"\"#F*!\"\"*&F'F0-F,6#,&F*F*F'F*F1" }{TEXT -1 51 "\n \n4. Find the area of the surface parametrized by " }{XPPEDIT 18 0 "f (t) = [t*cos(theta), t*sin(theta), t^2];" "6#/-%\"fG6#%\"tG7%*&F'\"\" \"-%$cosG6#%&thetaGF**&F'F*-%$sinG6#F.F**$F'\"\"#" }{TEXT -1 117 ", an d \nenclosed in the unit sphere centered at the origin.\n\n5.\011Find \+ the surface area of the surface parameterized by:" }{XPPEDIT 18 0 "[(p hi+2*cos(theta))*cos(phi), (phi+2*cos(theta))*sin(phi), 2*sin(theta)]; " "6#7%*&,&%$phiG\"\"\"*&\"\"#F'-%$cosG6#%&thetaGF'F'F'-F+6#F&F'*&,&F& F'*&F)F'-F+6#F-F'F'F'-%$sinG6#F&F'*&F)F'-F66#F-F'" }}{PARA 0 "" 0 "" {TEXT -1 4 " " }{XPPEDIT 18 0 "phi = Pi .. 4*Pi;" "6#/%$phiG;%#PiG* &\"\"%\"\"\"F&F)" }{TEXT -1 3 ", " }{XPPEDIT 18 0 "theta = 0 .. 2*Pi; " "6#/%&thetaG;\"\"!*&\"\"#\"\"\"%#PiGF)" }{TEXT -1 3 "\n \n" }}}} {MARK "0 0 2" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }