{VERSION 3 0 "IBM INTEL NT" "3.0" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "2D Comment" 2 18 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "2D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE " " -1 256 "Helvetica" 1 14 128 0 0 1 0 0 2 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 24 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "Helvetica " 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "Helvetica" 0 1 128 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 260 "Helvetica" 1 14 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "Helvetica" 1 14 0 0 255 1 0 0 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Map le Output" 0 11 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "Maple Plot" 0 13 1 {CSTYLE "" -1 -1 "" 0 1 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 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "" 18 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT 257 27 "Double and Triple Integ rals" }}{PARA 0 "" 0 "" {TEXT 256 17 "Mth 254 Fall 1999" }}{PARA 0 "" 0 "" {TEXT 258 16 "Bent E. Petersen" }}{PARA 0 "" 0 "" {TEXT 259 30 "F ilename: double_integrals.mws" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 249 "This worksheet illustrates Maple's limit ed support for double and triple integrals. You can get Maple to do qu ite a bit, but you will need to understand what you are doing, and you may need to do some of the work (for example, a change of variable). " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 141 "Maple support for double and triple integrals (via iterated integ rals) is part of the student package, so we need to load that package \+ first." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "with(student);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#7F%\"DG%%DiffG%*Dou bleintG%$IntG%&LimitG%(LineintG%(ProductG%$SumG%*TripleintG%*changevar G%(combineG%/completesquareG%)distanceG%'equateG%(extremaG%*integrandG %*interceptG%)intpartsG%(isolateG%(leftboxG%(leftsumG%)makeprocG%)maxi mizeG%*middleboxG%*middlesumG%)midpointG%)minimizeG%(powsubsG%)rightbo xG%)rightsumG%,showtangentG%(simpsonG%&slopeG%(summandG%*trapezoidG%&v alueG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "with(plots): # nee ded for implicitplot() below" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 194 "Let us look at the evaluation of a sim ple double integral. Note it is probably easier to do it by hand, but \+ this example illustrates some of what you will need to go through in t he general case." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 260 7 "Example" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 33 "Ev aluate the double integral of " }{XPPEDIT 18 0 "xy;" "6#%#xyG" } {TEXT -1 39 " over the region bounded by the line " }{XPPEDIT 18 0 " y = x-1;" "6#/%\"yG,&%\"xG\"\"\"\"\"\"!\"\"" }{TEXT -1 20 " and the p arabola " }{XPPEDIT 18 0 "y^2 = 2*x+6;" "6#/*$%\"yG\"\"#,&*&\"\"#\"\" \"%\"xGF*F*\"\"'F*" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 41 "First we find the points of intersection: " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "solve(\{y=x-1,y^2=2*x+6\},\{x,y\});" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$<$/%\"xG!\"\"/%\"yG!\"#<$/F(\"\"%/F%\"\"&" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 65 "No w we plot the curves over a slightly larger range (experiment):" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "implicitplot(\{y=x-1,y^2=2*x+6\},x=-4..6,y=-3..5,thickness=2);" }}{PARA 13 "" 1 "" {GLPLOT2D 400 300 300 {PLOTDATA 2 "6&-%'CURVESG6_q7 $7$$!1+++++++G!#:$!1(***********>h!#;7$$!1+++++?.HF*$!1-++++++WF-7$F.7 $$!1\\;s!p*)H$HF*$!14oAuC3OLF-7$7$$!1+++++!G*HF*$!1-++++++7F-F47$F:7$$ !1cjU0<#o)HF*$\"1%[3TVOd%H!#<7$7$$!1++++++!)HF*$\"1)*************>F-F@ 7$FG7$$!1I]z_:vRHF*$\"1X-OAC,=JF-7$7$$!1+++++!['GF*$\"1)************>& F-FM7$FS7$$!1)4<>\"*o>%GF*$\"1&yO`H^d`&F-7$7$F($\"1&eqk6FF*$!1-++++++wF-Fho7$F^pF'7 $Fin7$$!1_+Vpi.,FF*$\"18/Wb,H3wF-7$7$$!1+++++?ZEF*$\"1*************R)F -Fep7$F[q7$$!1LLLLLLPDF*$\"1kmmmmm)\\*F-7$7$F^o$\"1+++++?(3\"F*Faq7$7$ $!\"#\"\"!$!170#G^?GT\"F*7$$!1************>?F*$!1+++++++9F*7$FarF]o7$F gq7$$!1bbbbbbfBF*$\"1WWWWWkF6F*7$7$$!1**********>FBF*$\"1++++++g6F*Fhr 7$F^s7$$!1&fr)oR9f@F*$\"1ws4v^J(G\"F*7$7$F\\r$\"1)yyyyyyS\"F*Fds7$F[r7 $$!1\"o4(Q[N>>F*$!1bA.Hh^k9F*7$7$$!1,++++++;F*$!1p2Bp2Bp;F*F^t7$Fjs7$$ !1vW0)\\'p`>F*$\"1!eV%)>dHW\"F*7$7$$!1+++++![!>F*$\"1++++++![\"F*Fjt7$ F`u7$$!1.Y.5R$=t\"F*$\"1#oF!GrY&e\"F*7$7$Fet$\"1&eO&eO&em\"F*Ffu7$7$$! 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The limits of integration are found by solving for " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 13 " in terms of " }{XPPEDIT 18 0 "y;" " 6#%\"yG" }{TEXT -1 65 ". That is pretty trivial here, but let's let M aple do it for us:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "g1:=solve(y^2=2*x+6,x);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%#g1G,&*$)%\"yG\"\"#\"\"\"#\"\"\"F)!\"$F," }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "g2:=solve(y=x-1,x);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#g2G,&%\"yG\"\"\"F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "Doubleint(x*y,x=g1..g2,y=-2..4): % \+ = value(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-F%6$*&%\"xG \"\"\"%\"yGF+/F*;,&*$)F,\"\"#\"\"\"#F+F2!\"$F+,&F,F+F+F+/F,;!\"#\"\"% \"#O" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 125 "Note Doubleint() always returns unevaluated. You have to apply value() in order to get Maple to evaluate the double integral." }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 240 "When we \+ do interated integrals by hand we sometimes find it convenient, or eve n necessary, to change the order of integration. Sometimes Maple succe eds where we would fail, but returns an answer that is difficult to in terpret - for example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "Doubleint(x^3*exp(x^5),x=y..1,y=0.. 1): % = value(%); lhs(%)=evalf(lhs(%),28);" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-F%6$*&)%\"xG\"\"$\"\"\"-%$expG6#*$)F+\"\"&F- \"\"\"/F+;%\"yGF4/F7;\"\"!F4-%$intG6$,$*&,**(-%&GAMMAG6##\"\"%F3F4)!\" \"#F4F3F-),$*$)F7F3F-FHFEF-F4*(-FC6$FEFHF4FGF-FJF-FH*&)F7FFF-FBF-F4*&F RF--FC6$FEFKF4FHF-*$)FK#\"\"%F3F-!\"\"#FHF3F8" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-F%6$*&)%\"xG\"\"$\"\"\"-%$expG6#*$)F+\"\"&F- \"\"\"/F+;%\"yGF4/F7;\"\"!F4$\"=V\\d?2Z!4=pljlV$!#G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 155 "Note when we a pply evalf() to a Doubleint() Maple will employ a numeric quadrature m ethod. Thus we can (almost) always obtain an approximate numeric value ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 113 "If \+ we were doing this integral by hand we would have changed the order of integration first. Let's try that here:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "Doubleint(x^3*exp(x^ 5),y=0..x,x=0..1): % = value(%); lhs(%) = evalf(lhs(%),28);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-F%6$*&)%\"xG\"\"$\"\"\"-%$expG6#* $)F+\"\"&F-\"\"\"/%\"yG;\"\"!F+/F+;F8F4,&-F/6#F4#F4F3#!\"\"F3F4" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-F%6$*&)%\"xG\"\"$\"\"\"-%$e xpG6#*$)F+\"\"&F-\"\"\"/%\"yG;\"\"!F+/F+;F8F4$\"=V\\d?2Z!4=pljlV$!#G" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "Obviously Maple found the change of order of integration helpful too. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 261 7 "Example" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "Now let's conside r a triple integral (and a famous result of Isaac Newton). For a ball \+ B of radius " }{XPPEDIT 18 0 "a;" "6#%\"aG" }{TEXT -1 14 " with densit y " }{XPPEDIT 18 0 "delta;" "6#%&deltaG" }{TEXT -1 29 " the total mass M is given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 32 "Tripleint(delta(x,y,z),x,y,z,B);" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$-F$6$-%&deltaG6%%\"xG%\"yG%\"zG/F- ;%\"BG%!G/F.;F3F3/F/F5" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 82 "Let's place the ball so it's centered at \+ the origin and let's suppose the density " }{XPPEDIT 18 0 "delta;" "6# %&deltaG" }{TEXT -1 30 " depends only on the distance " }{XPPEDIT 18 0 "rho;" "6#%$rhoG" }{TEXT -1 49 " from the center. Then in spherical \+ coordinates (" }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 14 " = l ongitude, " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 22 " = colatitu de) we have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "Tripleint(delta(rho)*rho^2*sin(phi),rho=0..a,phi=0 ..Pi,theta=-Pi..Pi); % = value(%); M:=rhs(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$-F$6$*(-%&deltaG6#%$rhoG\"\"\")F.\"\"#\" \"\"-%$sinG6#%$phiGF//F.;\"\"!%\"aG/F6;F9%#PiG/%&thetaG;,$F=!\"\"F=" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-F%6$-F%6$*(-%&deltaG6#%$rh oG\"\"\")F/\"\"#\"\"\"-%$sinG6#%$phiGF0/F/;\"\"!%\"aG/F7;F:%#PiG/%&the taG;,$F>!\"\"F>,$*&-%$intG6$*&F,F3F1F3F8F0F>F0\"\"%" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"MG,$*&-%$intG6$*&-%&deltaG6#%$rhoG\"\"\")F.\"\"# \"\"\"/F.;\"\"!%\"aGF/%#PiGF/\"\"%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 79 "The z-component of the gravitat ional force of the ball on a point mass of mass " }{XPPEDIT 18 0 "m;" "6#%\"mG" }{TEXT -1 27 " located at the point (0,0," }{XPPEDIT 18 0 "s ;" "6#%\"sG" }{TEXT -1 9 "), where " }{XPPEDIT 18 0 "a < s;" "6#2%\"aG %\"sG" }{TEXT -1 13 ", is given by" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 154 "Tripleint((G*m*delta(rho)* rho^2*sin(phi)*(s-rho*cos(phi)))/(s^2-2*s*rho*cos(phi)+rho^2)^(3/2),ph i=0..Pi,theta=-Pi..Pi,rho=0..a); % = value(%); F:=-rhs(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%$IntG6$-F$6$-F$6$*&*.%\"GG\"\"\"%\"mGF--%& deltaG6#%$rhoGF-)F2\"\"#\"\"\"-%$sinG6#%$phiGF-,&%\"sGF-*&F2F--%$cosGF 8F-!\"\"F-F5*$),(*$)F;F4F5F-*(F;F-F2F5F=F5!\"#*$F3F5F-#\"\"$F4F5!\"\"/ F9;\"\"!%#PiG/%&thetaG;,$FNF?FN/F2;FM%\"aG" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#/-%$IntG6$-F%6$-F%6$*&*.%\"GG\"\"\"%\"mGF.-%&deltaG6#%$ rhoGF.)F3\"\"#\"\"\"-%$sinG6#%$phiGF.,&%\"sGF.*&F3F.-%$cosGF9F.!\"\"F. F6*$),(*$)FF6!\"#*$F4F6F.#\"\"$F5F6!\"\"/F:;\"\"!%# PiG/%&thetaG;,$FOF@FO/F3;FN%\"aG-%$intG6$,$*&*.F-F6F/F6F0F6F4F6,**&-%% sqrtG6#*$),&F3F.F%\"FG,$-%$intG6$,$*&*.%\"GG\"\"\" %\"mGF--%&deltaG6#%$rhoGF-)F2\"\"#\"\"\",**&-%%sqrtG6#*$),&F2F-%\"sG! \"\"F4F5F5F2F-F-*&F8F5F>F-F-*&-F96#*$),&F2F-F>F-F4F5F5F2F5F?*&FBF5F>F5 F-F-%#PiGF-F5*()F>\"\"#F5-F96#FDF5-F96#F;F5!\"\"F4/F2;\"\"!%\"aGF?" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "F:=simplify(F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"FG,$*&**%\"GG\"\"\"%\"mGF)%#PiGF)-%$intG 6$*,-%&deltaG6#%$rhoGF))F3\"\"#\"\"\",&-%%csgnG6#,&F3F)%\"sG!\"\"F=-F9 6#,&F3F)FF)F8F)/F3;\"\"!%\"aGF)F6*$)F<\"\"#F6!\"\"F5" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 470 "W e would like to get rid of the complex signum functions. We should be \+ able to do so by making use of the assume facility, but it seems trick y. We will save ourselves a headache by making substitutions for the \+ troublesome expressions (since we know their values). We are making a \+ simple syntactic substitution here - an expression is replaced by anot her expression with no reference to the value of either one - so care \+ is needed. (Actually, to be honest, it's a hack.)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "F:=subs(csg n(-s+rho)=-1,csgn(rho+s)=1,F);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\" FG,$*&**%\"GG\"\"\"%\"mGF)%#PiGF)-%$intG6$,$*&-%&deltaG6#%$rhoGF))F4\" \"#\"\"\"!\"#/F4;\"\"!%\"aGF)F7*$)%\"sG\"\"#F7!\"\"F6" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 305 "Because \+ M already has a value assigned to it we have to unassign it in order t o subtitute it into the formula for F. It is easy to confuse Maple her e by doing something nonsensical - so we have to be careful. Next to u se F in a formula we also need to unevaluate it, else it will be repla ced by its value." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 44 "N:=M: M:='M': simplify(F*M/N): F:='F': F=%%; " }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"FG,$*&*(%\"GG\" \"\"%\"mGF)%\"MGF)\"\"\"*$)%\"sG\"\"#F,!\"\"!\"\"" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 185 "We have derived N ewton's result that a ball with density depending only on the distance from the center attracts a point mass as if the mass of the ball were concentrated at the center." }}}}{MARK "29 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }