{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 "" -1 256 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 258 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 259 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 260 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 262 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 264 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 266 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "" 1 12 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 268 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 269 "" 0 1 0 0 0 0 0 1 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 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 } {CSTYLE "" -1 272 "" 0 1 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 }{CSTYLE "" -1 274 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 275 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 276 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 277 "" 0 1 0 0 0 0 1 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 280 "" 1 14 0 0 0 0 0 1 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 "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 8 4 0 0 0 0 0 0 -1 0 }{PSTYLE "H eading 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 8 2 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 "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 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 "" 0 257 1 {CSTYLE "" -1 -1 "" 1 10 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 "" 0 258 1 {CSTYLE "" -1 -1 "" 1 10 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 "" 0 259 1 {CSTYLE "" -1 -1 "" 1 10 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 {EXCHG {PARA 18 "" 0 "" {TEXT -1 47 "A Selection of Maple Illu strations for Calculus" }}{PARA 19 "" 0 "" {TEXT 279 13 "J. A. Ziegler " }}{PARA 256 "" 0 "" {TEXT -1 0 "" }}{PARA 257 "" 0 "" {TEXT -1 37 "S outhern Polytechnic State University" }}{PARA 259 "" 0 "" {TEXT -1 17 "jziegler@spsu.edu" }}{PARA 258 "" 0 "" {TEXT 278 44 "http://www2.SPSU .edu/math/ziegler/index.html" }}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 11 " Calculus I" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 67 " Does a Function U niquely Determine the Tangent Lines of its Graph?" }}{EXCHG {PARA 0 " " 0 "" {TEXT -1 116 "In many cases, each point on a curve is associate d with an unique tangent line. For example, consider the graph of " } {XPPEDIT 18 0 "y = 2-x^2/2;" "6#/%\"yG,&\"\"#\"\"\"*&%\"xG\"\"#\"\"#! \"\"F," }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "r estart:with(plots):with(linalg):with(plottools):" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 42 "We definine our curve: here, the graph of " } {XPPEDIT 18 0 "y = 2-x^2/2;" "6#/%\"yG,&\"\"#\"\"\"*&%\"xG\"\"#\"\"#! \"\"F," }{TEXT -1 36 ", using the parameterization x = t, " }{XPPEDIT 18 0 "y = 2-t^2/2;" "6#/%\"yG,&\"\"#\"\"\"*&%\"tG\"\"#\"\"#!\"\"F," } {TEXT -1 68 " for t in [-2, 2]. We represent C by the vector valued f unction r. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "r:=t->[t,2-t ^2/2]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "We use this to draw C. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "C:=plot([r(t)[1],r(t)[2 ],t=-2..2],color=blue,thickness=3):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 79 "We find the derivative of r(t) and use it to find the unit tang ent vector at t." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "dr:=una pply(diff(r(t),t),t):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "T: =unapply(dr(t)/sqrt(innerprod(dr(t),dr(t))),t):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 241 "We find the length of the arc along C from t = -1 t o t. This is approximated numerically as arc length integrals are not oriously difficult to evaluate in closed form. As a matter of general procedure, this permits a very wide choice for C." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "s:=t->evalf(Int(innerprod(dr(w),dr(w)),w=-1 ..t)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 293 "We calculate the arc l ength of C from t = -1 to t = 1. We use this as the length of the lin e segment which will represent the tangent line in the animation and, \+ below, use a smaller part of the corresponding arc of C as the collect ion of points with which the line segment will make contact. " }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "L:=s(1):" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 83 "We define the vector expression which we will use \+ to draw the tangent line segment." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "R:=evalm(r(t)+u*T(t)):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 38 "We draw the tangent line segment at t." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 63 "TL:=t->plot([R[1],R[2],u=-s(t)..L-s(t)],c olor=red,thickness=3):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "We mark the point of contact at t." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "P:=t->disk(r(t),0.05,color=green):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 146 "We define a sequence of values for t so that, in the ani mation, the tangent line segment will oscillate over 0.8 of the arc fr om t = -1 to t = 1. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "S:= seq(0.8*sin(Pi*k/25),k=0..50):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 45 "We create the necessary sequence of pictures." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 59 "pic:=display(seq(display(TL(t),P(t)),t=S),inse quence=true):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "We display these using C as a background." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "display(pic,C,scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 65 "We may add the slope at each value of t to our picture. \+ Because " }{XPPEDIT 18 0 "dy/dx = [dy/dt]/[dx/dt];" "6#/*&%#dyG\"\"\"% #dxG!\"\"*&7#*&F%F&%#dtGF(F&7#*&F'F&F,F(F(" }{TEXT -1 11 " , we have \+ " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "m:=t->dr(t)[2]/dr(t)[1] :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 12 "Then we use " }{TEXT 274 8 " textplot" }{TEXT -1 55 " to display the value of the slope for each va lue of t." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "Tx:=t->textpl ot([0.1,3.1,cat(\"m = \",convert(evalf(m(t),3),string))],font=[HELVETI CA,BOLD,14],align=\{ABOVE,RIGHT\}):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "Again, we create the needed sequence of pictures and display t hem against the background provided by C." }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 66 "pic2:=display(seq(display(TL(t),P(t),Tx(t)),t=S),in sequence=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "display( pic2,C,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 172 "In this case, the tange nt line is clearly determined by the curve. This information is there fore contained in the analytic description of the curve: here, in the \+ equation " }}{PARA 0 "" 0 "" {TEXT -1 5 "y = " }{XPPEDIT 18 0 "2-x^2/ 2" "6#,&\"\"#\"\"\"*&%\"xG\"\"#\"\"#!\"\"F*" }{TEXT -1 5 ". " }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 147 "However, sometimes a curve has a \+ point which is not associated with an unique tangent line. For exampl e, consider the point (1,1) on the graph of " }{XPPEDIT 18 0 "y = -abs (x-1)+1;" "6#/%\"yG,&-%$absG6#,&%\"xG\"\"\"\"\"\"!\"\"F-\"\"\"F+" } {TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "restart:wi th(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "F:=x->-abs(x- 1)+1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "G:=plot(F(x),x=-1. .3,color=blue,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "L1:=t->plot([1+s*(3/2*cos(t)),1+s*(3/2*sin(t)),s=-1..1],color=red, thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 37 "T:=seq(Pi/ 6*sin(-2*Pi*k/24),k=0..24):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "Tx:=t->textplot([0.5,2,cat(\"m = \",convert(evalf(tan(t),3),stri ng))],font=[HELVETICA,BOLD,14],align=\{ABOVE,RIGHT\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 61 "pic2:=display(seq(display(Tx(t),L1( t)),t=T),insequence=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "display(G,pic2,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 175 "These illust rations may be used with the point-slope form of the equation of a lin e to motivate the association of an unique tangent line with the exist ance of the derivative." }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 21 " Swe eping Out an Area" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 199 " The app roximation of the \"area under a curve\" by a suitable Riemann sum imp roves as the norm of the partition is made smaller. Maple provides a \+ way of illustrating this through commands in the " }{TEXT 256 7 "stude nt" }{TEXT -1 99 " package. These commands are easy to use and, using the uniform partition, are suggestively named " }{TEXT 257 32 "leftbo x, rightbox, and middlebox" }{TEXT -1 30 ". The corresponding sums ar e " }{TEXT 258 18 "leftsum, rightsum," }{TEXT -1 5 " and " }{TEXT 259 9 "middlesum" }{TEXT -1 14 ". The use of " }{TEXT 260 8 "leftbox " } {TEXT -1 4 "and " }{TEXT 261 8 "leftsum " }{TEXT -1 83 "is illustrated first. As this is typical, the use of the others will be omitted. \+ " }}{PARA 0 "" 0 "" {TEXT -1 620 " However, fairly early in a fir st calculus course it becomes desirable to introduce, particularly to \+ engineering students, the idea of \"generating\" the area by \"sweepin g it out\" with a vertical line segment. Of course, this is only a gl oss on Leibniz's error of passing to the limit before summing and so i t makes many of us uneasy to teach it, but, faced with a multivariable calculus class of normal abilities, our principles may weaken and we \+ may find ouselves sweeping out volumes just as we did when we were stu dents. The single variable case will be illustrated second and higher dimensional cases later on." }}{PARA 0 "" 0 "" {TEXT -1 49 " We \+ begin by loading the necessary packages." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "restart:with(student):with(plots):with(plottools):" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 77 " Let us consider a classic case: the area \"under the curve\" for f(x) = " }{XPPEDIT 18 0 "x(3-x );" "6#-%\"xG6#,&\"\"$\"\"\"F$!\"\"" }{TEXT -1 24 " on the interval [0 , 2]." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f:=x->x*(2-x):" }} }{EXCHG {PARA 0 "" 0 "" {TEXT -1 80 "Choose the number of rectangles. \+ We begin with 10 and later change this to 100." }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 7 "n:=100;" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "LP:=leftbox (f(x),x=0..2,n,shading=yellow):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "LS:=leftsum(f(x),x=0..2,n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "LT:=textplot([1,1.1,cat(\" Approx. Area = \",convert( evalf(LS,5),string))],font=[HELVETICA,BOLD,14]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(LP,LT);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 594 "The idea of \"sweeping out the area\" comes quite natura lly, if we consider how we would carry out, by hand, the calculation o f an approximation involving n rectangles. Then it is natural to work from left to right, calculating the area of a rectangle and adding it to the partial sum already obtained. This way, each addition involve s only two terms and mistakes are less likely. As one works, an image forms in one's mind of the part of the work which has been accomplish ed as the geometrical equivalent of the partial sum to this point. We can make a Maple animation of this process using " }{TEXT 262 7 "left box" }{TEXT -1 1 " " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "a:=0;b:=2;del:=(b-a)/n;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "X:=k->a+k*del:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "r:=k->leftbox(f(x),x=0..X(k),k,colo r=blue,shading=yellow):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 32 " p:=plot(f(x),x=0..2,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "R:=seq(display(p,r(i)),i=1..n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "S:=i->evalf(sum(f(X(k))*del,k=0..i),5):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "S:=i->evalf(leftsum(f(x),x=0 ..X(i),i),5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "Tn:=i->te xtplot(\{[1,1.3,\"Area = 4/3\"],[1,1.1,cat(\" Sum = \",convert(S(i),st ring))]\},font=[HELVETICA,BOLD,14]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "RN:=seq(display(p,r(i),Tn(i)),i=1..n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "display(RN,insequence=true);" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 230 "Now, one would like to do away wi th the rectangles and just sweep out the area, leaving a solid region \+ of color behind. Here's how to do that. The value of the integral fr om 0 to the current positon of the red line is displayed." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "P:=t->plot(f(x),x=0..t,filled=true, color=COLOR(RGB,0.0,0.7,1.0)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 69 "L:=t->line([t,0],[t,f(t)],color=COLOR(RGB,0.73,0.0,0.0),thickn ess=4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "pic:=seq(display (L(t/75),P(t/75)),t=0..150):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "Tx:=t->textplot([1,1.2,cat(\" Integral = \",convert(evalf(int(f (x),x=0..t),4),string))],font=[HELVETICA,BOLD,14]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "picT:=seq(display(p,L(t/75),P(t/75),Tx(t/ 75)),t=0..150):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "display( picT,insequence=true);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 35 " Sweeping Out Volumes of Revo lution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 340 "We will illustrate find ing volumes of revolution by \"sweeping them out\" with thin disks or \+ cylindrical shells. This is a common \"engineering\" approach having \+ its origin in Leibniz's view of integration. As is well known, it is \+ helpful for setting up the necessary integrals and later in writing mu ltiple integrals as iterated integrals. " }{TEXT 263 75 "In each case , switch to maximum magnification before running the animation." }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "First, let us fill a volume of rev olution. We revolve " }{XPPEDIT 18 0 "y = 2*x-x^2;" "6#/%\"yG,&*&\"\" #\"\"\"%\"xGF(F(*$F)\"\"#!\"\"" }{TEXT -1 125 " about the x-axis on [0 , 2]. By changing R and the interval appropriately in the following c ode, other examples may be used." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:with(plottools):with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "R:=x->2*x-x^2:" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 92 "surf:=s->plot3d([x,R(x)*cos(t),R(x)*sin(t)], x=0..s,t=0..2*Pi,color=green,style=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "slice:=s->plot3d([s,r*cos(t),r*sin(t)],r=0..R (s),t=0..2*Pi,color=red,style=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "frame:=plot3d([x,R(x)*cos(t),R(x)*sin(t)],x=0..2, t=0..2*Pi,style=wireframe,color=black,grid=[15,20]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "S:=[seq(s/25+0.001,s=0..50)]:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Now let us watch the slice alone sweeping out the volume." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "pic1:=d isplay(seq(slice(s),s=S),insequence=true):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 73 "display(pic1,frame,scaling=constrained,axes=normal, orientation=[-70,70]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Next, w e fill the volume as the slice sweeps it out." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "pic:=display(seq(display(surf(s),slice(s)),s=S), insequence=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "displa y(pic,frame,scaling=constrained,axes=normal,orientation=[-70,70]);" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 88 "By calculating and displaying the volume at each step, on e may emphasise the connection." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "V1:=s->int(Pi*R(x)^2,x=0..s):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 114 "Text:=s->textplot3d([0,1,1.2,cat(`Volume =`,convert( V1(s),string))],align=ABOVE,font=[TIMES,BOLD,14],color=black):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "pic1:=display(seq(display(su rf(s),slice(s),Text(s-0.001)),s=S),insequence=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "display(pic1,frame,scaling=constrained,ax es=normal,orientation=[-74,76]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "The integral \+ for the volume is " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "Int(P i*R(x)^2,x=0..2)=int(Pi*R(x)^2,x=0..2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Approximately" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "evalf(Int(Pi*R(x)^2,x=0..2),4);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Next, we fill a volume of revolution using cylindrical shells. \+ We revolve " }{XPPEDIT 18 0 "y = 2*x-x^2" "6#/%\"yG,&*&\"\"#\"\"\"%\" xGF(F(*$F)\"\"#!\"\"" }{TEXT -1 28 " on [0, 2] about the y-axis." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "Frame:=plot3d([r*cos(t),r*si n(t),R(r)],r=0..2,t=0..2*Pi,style=wireframe,color=black,grid=[13,20]): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "shell:=r->plot3d([r*cos (t),r*sin(t),z],t=0..2*Pi,z=0..R(r),color=yellow,style=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "fill:=r->plot3d([s*cos(t) ,s*sin(t),R(s)],s=0..r,t=0..2*Pi,color=red,style=patchnogrid):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "bottom:=r->plot3d([s*cos(t), s*sin(t),0],s=0..r,t=0..2*Pi,color=red,style=patchnogrid):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 57 "Again, we first let the shell alone sweep out the volume." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "pic2:=d isplay(seq(shell(r),r=S),insequence=true):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 71 "display(pic2,Frame,axes=normal,scaling=constrained, orientation=[0,60]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "pic 3:=display(seq(display(bottom(r),shell(r),fill(r)),r=S),insequence=tru e):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "display(pic3,Frame,a xes=normal,scaling=constrained,orientation=[0,60]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 72 "A gain, we calculate the volume at each step and add this to our picture ." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "V2:=r->evalf(Int(2*Pi* x*R(x),x=0..r),4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 115 "Text 2:=s->textplot3d([0,0,1.4,cat(`Volume =`,convert(V2(s),string))],align =ABOVE,font=[TIMES,BOLD,14],color=black):" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 91 "pic3:=display(seq(display(bottom(r),shell(r),fill(r ),Text2(r-0.001)),r=S),insequence=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "display(pic3,Frame,axes=normal,scaling=constrained,or ientation=[0,80]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 32 "The integra l for this volume is " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "In t(2*Pi*x*R(x),x=0..2)=int(2*Pi*x*R(x),x=0..2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 13 "Approximately" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 33 "evalf(Int(2*Pi*x*R(x),x=0..2),4);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 13 " Calculu s III" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 80 " Volumes by Double Integ ration: Drawing the Object and Animating the Integration" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:with(plots):with(plottools) :" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 573 "In order to find the volume between the x,y-plane and a surface z = f(x,y) over a closed, bounded region, R, of the x,y-plane it usually necessary to visualize it. To see how to use use Maple to help do this, it is well to begin with a \+ simple case in which R is vertically simple or horizontally simple. L ines and spacecurves to draw R, then only that portion of the surface \+ above (or below) R will be drawn, and, finally the vertices of these w ill be connected with line segments. It will be convenient to define \+ f as a Maple function. Consider the following example. " }}}{EXCHG {PARA 4 "" 0 "" {TEXT 267 37 "Example 1: A Vertically Simple Region" } }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 69 "Find the volume enclosed between the graph of the function f(x,y) = " }{XPPEDIT 18 0 "4-(x/2)^2-(y/3) ^2;" "6#,(\"\"%\"\"\"*$*&%\"xGF%\"\"#!\"\"\"\"#F**$*&%\"yGF%\"\"$F*\" \"#F*" }{TEXT -1 83 " and the x,y-plane, over the region R enclosed by the curves x = 0, y = x + 3, and " }}{PARA 0 "" 0 "" {TEXT -1 4 "y = \+ " }{XPPEDIT 18 0 "x^2+1;" "6#,&*$%\"xG\"\"#\"\"\"\"\"\"F'" }{TEXT -1 58 ". Here R is vertically simple. The \"lower curve\" is y = " } {XPPEDIT 18 0 "x^2+1;" "6#,&*$%\"xG\"\"#\"\"\"\"\"\"F'" }{TEXT -1 36 " and the \"upper curve\" is y = x + 3." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "First, define the function. The parentheses in (x,y) are important." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f:=(x,y)->4- (x/2)^2-(y/3)^2:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "Next, define \+ the upper and lower curves as functions." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "Uc:=x->x+3:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "Lc:=x->x^2+1:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "solv e(Uc(x)=Lc(x),x);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "So, in R, x will be in [0,2]. Now use this fact and the upper and lower curve fu nctions to draw the graph of the surface over R." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "s:=plot3d(f(x,y),x=0..2,y=Lc(x)..Uc(x),grid=[ 15,15]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "Now, draw the region R. If the lower and upper curves did not meet, as they do in this exa mple, there would be a second line at the right-hand edge of R." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "L1:=line([0,1,0],[0,3,0],col or=blue,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "L2 :=spacecurve([x,Uc(x),0],x=0..2,color=blue,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "L3:=spacecurve([x,Lc(x),0],x=0..2,c olor=blue,thickness=2):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 125 "Colle ct these together and call the result R. In order to show R in its ex pected relationship to the coordinate system, the " }{TEXT 264 4 "view " }{TEXT -1 38 " command has been used in the display." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "R:=\{L1,L2,L3\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "display(s,R,axes=normal,view=[0..2,0..5,0 ..4]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 92 "Now define the vertical connecting lines between the vertices of R and those of the surface. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "C1:=line([0,1,0],[0,1,f (0,1)],color=green,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "C2:=line([0,3,0],[0,3,f(0,3)],color=green,thickness=3 ):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "C3:=line([2,5,0],[2,5 ,f(2,5)],color=green,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "C:=\{C1,C2,C3\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "display(s,R,C,axes=normal,view=[0..2,0..5,0..4]);" } {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 161 "Rotating the pic ture helps to reveal completely the volume that will be given by the i terated integral, taken first with respect to y and then with respect \+ to x:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "V:=Int(Int(f(x,y), y=Lc(x)..Uc(x)),x=0..2);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 29 "The v alue of this is given by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 " value(V);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 46 "Of course, its value may be found directly via" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "int(int(f(x,y),y=Lc(x)..Uc(x)),x=0..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 50 "Animating \+ the Process by \"Sweeping Out\" the Volume" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 461 "To show the outer integral \"s weeping out the volume\", one may use a parametric surface to represen t the \"very thin\" slice of the volume at x. The Maple function prod ucing this slice is is an \"object-valued\" function of x and the para meters are y and z. Its definition assumes that the surface is above \+ the x,y-plane over all of R. However, it succeeds equally well when t he situation is reversed. To show the idea, first a single slice is \+ drawn at x = 1. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "A:=x-> plot3d([x,y,z],y=Lc(x)..Uc(x),z=0..f(x,y),color=red,style=patchnogrid) :" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "display(A(1),s,R,C,axe s=normal,view=[0..2,0..5,0..4],style=wireframe);" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 343 "First, however, let us show the inner integral \" sweeping out\" a \"slice\" in the y-direction at a particular value of x. We make use of the ideas developed in sweeping out the area in tw o dimensions and modify the parametric surface we have just seen by dr awing it from the lower curve to y = t, where t lies between the lower and upper curves." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "A1:=( x,t)->plot3d([x,y,z],y=Lc(x)..t,z=0..f(x,y),color=red,style=patchnogri d):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "As in two dimensions, we \+ define a vertical line segment at (x,t) which will seem to sweep out t he area of the slice." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "vL :=(x,t)->line([x,t,0],[x,t,f(x,t)],color=black,thickness=3):" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "For x = 0.5, t = 3 we have" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "display(A1(0.5,3),vL(0.5,3), s,R,C,axes=normal,view=[0..2,0..5,0..4],style=wireframe);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 112 "We now use the objects to animate the in ner integral for any desired value of x and show the result for x = 0. 5." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 130 "pic4:=x->display(seq (display(A1(x,0.001+Lc(x)+s*(Uc(x)-Lc(x))/25),vL(x,0.001+Lc(x)+s*(Uc(x )-Lc(x))/25)),s=0..25),insequence=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 91 "display(pic4(0.5),s,R,C,vL(0.5,Lc(0.5)),axes=normal,v iew=[0..2,0..5,0..4],style=wireframe);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 129 "To make an a nimation of the \"slice\" sweeping out of this volume, a sequence of p ictures is made, then displayed using the option " }{TEXT 265 15 "inse quence=true" }{TEXT -1 197 ". By using x = i/20 +0.001 as the argumen t of A, a sequence of 40 pictures as x varies from 0 to 20 can be made and, because of the 0.001 term, the plotting error for i = 40 can be \+ avoided. The " }{TEXT 266 11 "orientation" }{TEXT -1 212 " option ca n initially be omitted as the point of view the animation can best be \+ seen is, of course, not yet known. To run the animation, click on the \+ picture and use the buttons which appear on the \"context bar\"." }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "SV:=display(seq(A(i/20+0.001 ),i=0..40),insequence=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "display(SV,s,C,R,axes=normal,view=[0..2,0..5,0..4],orientation=[ -23,56],style=wireframe);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 20 "Generating the Solid" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 345 "This part is new and is a good il lustration of the use of \"graphical object valued\" Maple functions. \+ (These are an invention of my own, although very likely many others w orking with Maple have invented them, too.) The 0.001 in the function al arguments prevent the plotting commands from producing what would b e essentially \"empty plot\" errors." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "s1:=plot3d(f(x,y),x=0..2,y=Lc(x)..Uc(x),grid=[10,5],s tyle=wireframe,color=black):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "surf:=t->plot3d(f(x,y),x=0..t,y=Lc(x)..Uc(x),grid=[10,5],color=b lue,style=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 " bottom:=t->plot3d(0,x=0..t,y=Lc(x)..Uc(x),grid=[10,5],color=blue,style =patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "Lside:=t- >plot3d([x,Lc(x),z],x=0..t,z=0..f(x,Lc(x)),color=yellow,style=patchnog rid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "Uside:=t->plot3d([ x,Uc(x),z],x=0..t,z=0..f(x,Uc(x)),color=yellow,style=patchnogrid):" }} }{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "LLine:=t->line([t,Lc(t),0], [t,Lc(t),f(t,Lc(t))],color=black,thickness=3):" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 73 "ULine:=t->line([t,Uc(t),0],[t,Uc(t),f(t,Uc(t)) ],color=black,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "Bline:=t->line([t,Lc(t),0],[t,Uc(t),0],color=black,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 179 "SV2:=seq(display(A(0),A( i/20+0.001),surf(i/20+0.001),bottom(i/20+0.001),Lside(i/20+0.001),Usid e(i/20+0.001),LLine(i/20+0.001),ULine(i/20+0.001),Bline(i/20+0.001),s1 ,C,R),i=0..40):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "display( SV2,insequence=true,axes=normal,view=[0..2,0..5,0..4],orientation=[-23 ,56]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 56 " A Volume in Spherical Coordinates by Tri ple Integration" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:w ith(plots):with(plottools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" } {TEXT 273 69 "Finding the volume using iterated integrals in spherical coordinates." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 47 "Find the volume \+ enclosed below by the cone z = " }{XPPEDIT 18 0 "sqrt(x^2+y^2);" "6#-% %sqrtG6#,&*$%\"xG\"\"#\"\"\"*$%\"yG\"\"#F*" }{TEXT -1 33 " and above b y the hemisphere z = " }{XPPEDIT 18 0 "sqrt(9-x^2-y^2);" "6#-%%sqrtG6# ,(\"\"*\"\"\"*$%\"xG\"\"#!\"\"*$%\"yG\"\"#F," }{TEXT -1 33 " and illus trate this graphically." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 269 48 "Visua lization of a possible order of integration" }}}{EXCHG {PARA 0 "" 0 " " {TEXT -1 157 "Note that when y = 0 and x > 0, the first equation red uces to z = x. Likewise, when x = 0 and y > 0, z = y. Thinking in sp herical coordinates, we see that " }{XPPEDIT 18 0 "phi = Pi/4;" "6#/%$ phiG*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 116 " at the wall of the cone. \+ So the portion of the hemisphere we need may be drawn in the followin g way. By choosing " }{TEXT 272 18 "style = wireframe," }{TEXT -1 71 " we will be able to see into the solid. It is convenient to use t fo r " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 11 " and p for " } {XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 1 "," }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 90 "S:=plot3d(3,t=0..2*Pi,p=0..Pi/4,coords=spheric al,style=wireframe,color=black,grid=[25,5]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "c:=plot3d([rho,t,Pi/4],rho=0..3,t=0..2*Pi,coord s=spherical,style=wireframe,color=black,grid=[5,25]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "display(S,c,scaling=constrained,ori entation=[-25,75]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Let us add a representative radius line to our picture." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "R:=spacecurve([r,0,Pi/6],r=0..3,coords=spherical ,color=red,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "display(S,c,R,scaling=constrained,orientation=[-25,75]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "It is clear that, inside our volume, " }{XPPEDIT 18 0 "rho;" "6 #%$rhoG" }{TEXT -1 39 " may run from 0 to 3, independently of " } {XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 8 ", while " }{XPPEDIT 18 0 "phi;" "6#%$ phiG" }{TEXT -1 16 " runs from 0 to " }{XPPEDIT 18 0 "pi/4;" "6#*&%#pi G\"\"\"\"\"%!\"\"" }{TEXT -1 20 " , independently of " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 93 ". Having realized this, we can no w write the necessary integrals. Integrating in the order " } {XPPEDIT 18 0 "rho,phi,theta;" "6%%$rhoG%$phiG%&thetaG" }{TEXT -1 53 " , and remembering that in spherical coordinates dV = " }{XPPEDIT 18 0 "rho^2;" "6#*$%$rhoG\"\"#" }{XPPEDIT 18 0 "sin(phi);" "6#-%$sinG6#%$ph iG" }{XPPEDIT 18 0 "d*rho;" "6#*&%\"dG\"\"\"%$rhoGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d*phi;" "6#*&%\"dG\"\"\"%$phiGF%" }{TEXT -1 1 " " } {XPPEDIT 18 0 "d*theta;" "6#*&%\"dG\"\"\"%&thetaGF%" }{TEXT -1 10 ", w e have " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "V:=Int(Int(Int(r ho^2*sin(phi),rho=0..3),phi=0..Pi/4),theta=0..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(V);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 " " }{TEXT 270 39 "Illustrating the process of integration" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 26 "The first integration, of " }{XPPEDIT 18 0 "rho;" "6#%$rhoG" }{TEXT -1 27 " from 0 to 3 while holding " } {XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "th eta;" "6#%&thetaG" }{TEXT -1 130 " fixed, produces the line segment we have already seen. We can illustrate this with an animation. Use th e maximum magnification." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "R1:=t->spacecurve([r,0,0],r=0..t,coords=spherical,color=red,thickness =3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "pic1:=display(seq(R 1(0.001+s/3),s=0..9),insequence=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "display(pic1,S,c);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 27 "The second in tegration, of " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }{TEXT -1 11 " from 0 to " }{XPPEDIT 18 0 "Pi/4;" "6#*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT -1 15 " while holding " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" }{TEXT -1 213 " fixed produces a wedge-shaped surface. We may show this surface by u sing the following parametric surface, which we define as a function o f s in a way that will be useful in a moment. We abbreviate \"phi\" b y p. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "s1:=s->plot3d([rh o,2*Pi*s/60,p],rho=0..3,p=0..Pi/4,coords=spherical,color=red,style=pat chcontour): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "display(S, c,s1(0),scaling=constrained,orientation=[-25,75]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 20 "We \+ now animate this." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "s2:=( s,t)->plot3d([rho,2*Pi*s/60,p],rho=0..3,p=0..t,coords=spherical,color= red,style=patchcontour): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 64 "pic2:=display(seq(s2(0,0.001+k*Pi/40),k=0..10),insequence=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "display(R1(3),S,c,pic2,sc aling=constrained,orientation=[-57,75]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "Now we make \+ use of our \"surface-valued\" function, s1, to make an animation showi ng the surface \"sweeping out\" the volume as " }{XPPEDIT 18 0 "theta; " "6#%&thetaG" }{TEXT -1 16 " runs from 0 to " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"\"\"%#PiGF%" }{TEXT -1 80 ". Change the magnification to \+ its maximum setting before running the animation." }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 37 "pic:=seq(display(S,c,s1(t)),t=0..59):" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "display(pic,insequence=true, scaling=constrained,orientation=[-25,75]);" }}}{EXCHG {PARA 0 "> " 0 " " {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 271 53 "A more elaborate illustration of the last integration" }}} {EXCHG {PARA 0 "" 0 "" {TEXT -1 203 "By redefining the top and bottom \+ surfaces as \"surface valued\" functions, we may make an animation tha t really generates the volume. The arguments \"s + 0.001\" in the fun ctions sa and ca are used to keep " }{TEXT 268 6 "plot3d" }{TEXT -1 157 " from generating an error when s = 0. The line, L, has been adde d to improve visibility. Again, set the maximum magnification before \+ running the animation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 89 "s a:=s->plot3d(3,t=0..2*Pi*s/60,p=0..Pi/4,coords=spherical,style=patchno grid,color=green):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "ca:=s ->plot3d([rho,t,Pi/4],rho=0..3,t=0..2*Pi*s/60,coords=spherical,style=p atchnogrid,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 " L:=line([0,0,0],[0,0,3],color=navy,thickness=4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "pic1:=seq(display(L,S,c,s1(0),s1(s),sa(s+0.00 1),ca(s+0.001)),s=0..60):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "display(pic1,insequence=true,scaling=constrained,orientation=[-25, 75]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 51 " Illustrating the Definition of the Doubl e Integral" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:with(p lots):with(plottools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 86 "To illu strate the definition of the double integral, let us use the function \+ f(x,y) = " }{XPPEDIT 18 0 "20-x^2-y^2;" "6#,(\"#?\"\"\"*$%\"xG\"\"#!\" \"*$%\"yG\"\"#F)" }{TEXT -1 33 " over the quarter disk of radius " } {XPPEDIT 18 0 "sqrt(20);" "6#-%%sqrtG6#\"#?" }{TEXT -1 43 " centered a t the origin and in quadrant I. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f:=(x,y)->20-x^2-y^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "P:=plot3d(f(x,y),x=0..sqrt(20),y=0..sqrt(20-x^2),styl e=patchnogrid,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "C:=spacecurve([sqrt(20)*sin(t),sqrt(20)*cos(t),0],t=0..Pi/2,thickn ess=4,color=navy):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "displ ay(P,C,axes=normal,orientation=[30,45]);" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 56 "Using the same number of subintervals in both directions " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 24 "To start, let us divide " } {XPPEDIT 18 0 "[0, sqrt(20)];" "6#7$\"\"!-%%sqrtG6#\"#?" }{TEXT -1 153 " into 10 equal subintervals in both the x- and y-directions. Af terwards, this should be increased to see the resulting refinement of \+ the approximation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a:=0: b:=sqrt(20):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "n:=10;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Delta:=(b-a)/n:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "X:=[seq(evalf(a+i*Delta),i=0..n)]: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "Y:=X:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 15 "Now we use the " }{TEXT 275 0 "" }{TEXT -1 0 " " }{TEXT 276 6 "cuboid" }{TEXT -1 11 ", from the " }{TEXT 277 10 "plot tools " }{TEXT -1 161 "package, to erect a rectangle on each of the gr id squares lying wholly within the quarter disk. The height is the va lue of f(x,y) at the midpoint of the square." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 279 "pic:=proc(X,Y)\n local L, k, j;\n L:=\{\}; \n for j from 1 to n do\n for k from 1 to n while f(X[k+1], Y[j+1])>0 do\n L:=L union \{cuboid([X[k],Y[j],0],[X[k+1],Y[j+1 ],f((X[k]+X[k+1])/2,(Y[j]+Y[j+1])/2)],color=red)\};\n \n \+ od;\n od;\n display(L);\nend:" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "We display the result, first without the surface, and then with it." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "display(pic(X,Y),C, axes=normal,orientation=[30,45]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "display(pic(X,Y),P,C,axes=normal,orientation=[30,45]) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 4 "" 0 "" {TEXT -1 67 "Using a different number of subintervals in the x- a nd y-directions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart: with(plots):with(plottools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f:=(x,y)->20-x^2-y^2:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 81 "P:=plot3d(f(x,y),x=0..sqrt(20),y=0..sqrt(20-x^2),style=patchnogrid ,color=yellow):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "C:=space curve([sqrt(20)*sin(t),sqrt(20)*cos(t),0],t=0..Pi/2,thickness=4,color= navy):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "a:=0:b:=sqrt(20): " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "We begin with 15 subinterval s in the x-direction and 10 in the y-direction. Again, these should l ater be increased." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "n:=15; " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "Delta:=(b-a)/n:" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "X:=[seq(evalf(a+i*Delta),i=0 ..n)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "c:=0:d:=sqrt(20): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "m:=10;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "DeltaY:=(d-c)/m:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "Y:=[seq(evalf(c+j*DeltaY),j=0..m)]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 279 "pic:=proc(X,Y)\n local L, k, j;\n L:=\{\};\n for j from 1 to m do\n for k from 1 to n wh ile f(X[k+1],Y[j+1])>0 do\n L:=L union \{cuboid([X[k],Y[j],0], [X[k+1],Y[j+1],f((X[k]+X[k+1])/2,(Y[j]+Y[j+1])/2)],color=red)\};\n \+ \n od;\n od;\n display(L);\nend:" }}}{EXCHG {PARA 0 " " 0 "" {TEXT -1 52 "Note that the \"blocks\" no longer have a square b ase." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 52 "display(pic(X,Y),C, axes=normal,orientation=[30,45]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "display(pic(X,Y),P,C,axes=normal,orientation=[30,45]) ;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 280 44 "http://www2.SPSU.edu/math/ziegler/in dex.html" }}}}}}{MARK "1" 0 }{VIEWOPTS 1 1 0 1 1 1803 }