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{SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT -1 51 "Selected Maple Graphics a
nd Animations for Calculus" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
256 "" 0 "" {TEXT -1 20 "J. A. Ziegler, Ph.D." }}{PARA 257 "" 0 "" 
{TEXT -1 37 "Southern Polytechnic State University" }}{PARA 258 "" 0 "
" {TEXT -1 17 "jziegler@spsu.edu" }}{PARA 259 "" 0 "" {TEXT 293 44 "ht
tp://www2.SPSU.edu/math/ziegler/index.html" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 282 "ABSTRACT:  Maple graphic
s and animations are presented which illustrate selected calculus topi
cs rranging from a line tangent to a curve to volumes in spherical coo
rdinates.  Topics include integration, volumes of revolution, and a pr
oblem associated with a multivarible chain rule." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 15 " DIFFERENTIATON
" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 52 "When the Slope is Determined \+
by the Point of Contact" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 354 "We define a \+
function, f, and the associated tangent line function, g, and create a
n animation in which a segment, of specified length, of the tangent li
ne at x = c is shown for each point in a specified interval.  The purp
ose is to illustrate the dependence of the slope on the point, as thou
gh one were touching a wooden model of the curve with a ruler." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:with(plots):with(plo
ttools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "F:=x->x*(3-x);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "l:=3/2;" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "r:=c->l/(sqrt(1+(D(F)(c))^2));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "m:=c->D(F)(c):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 84 "L:=c->line([c-r(c),-m(c)*r(c)+F(c)]
,[c+r(c),m(c)*r(c)+F(c)],thickness=3,color=blue):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 36 "p:=c->disk([c,F(c)],0.1,color=cyan):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 150 "s:=a->seq(a+1-cos(Pi*k/20),
k=0..40):# This specifies the interval [a,a+2] over which the point of
 contact                                       moves." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "C1:=plot(F(x),x=-.5..3.5,color=red,
thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "Tx:=c->te
xtplot([2,3.5,cat(\"m  = \",convert(evalf(D(F)(c),3),string))],font=[T
IMES,BOLD,14],align=\{ABOVE,RIGHT\}):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "P1:=seq(display(p(c),L(c),Tx(c)),c=s(.5)):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "pic1:=display(P1,insequence=
true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "display(pic1,C1,s
caling=constrained,view=[-2..5,-2..4]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 140 "However, som
etimes a curve has a point which is not associated with an unique slop
e.  For example, consider the point (1,1) on the graph of " }{XPPEDIT 
18 0 "y = -abs(x-1)+1;" "6#/%\"yG,&-%$absG6#,&%\"xG\"\"\"\"\"\"!\"\"F-
\"\"\"F+" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 
"restart:with(plots):with(plottools):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 18 "F:=x->-abs(x-1)+1:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "G:=plot(F(x),x=-1..3,color=blue,thickness=2):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "m:=k->k*Pi/30:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "h:=(3/2)/sqrt(1+m(k)^2):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "L:=m->line([1-h,-m(k)*h+1],[
1+h,m(k)*h+1],color=red,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 113 "Tx:=k->textplot([0.5,2,cat(\"m  = \",convert(evalf(m
(k),3),string))],font=[HELVETICA,BOLD,14],align=\{ABOVE,RIGHT\}):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 46 "fms:=seq(display(Tx(k),L(tan
(m(k)))),k=-5..5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "pic:=
display(fms,insequence=true):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 15 "display(G,pic);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "The Slope as a Limit" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "re
start:with(plots):with(plottools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }{TEXT 256 37 "We define our function, then plot it." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "f:=x->-x^3+2*x^2+8*x;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "p1:=plot(f(x),x=-3..4.5):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "display(p1);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 259 56 "We want to draw a line ta
ngent to this curve at x = 2.6." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 
"" }{TEXT 257 82 "We use calculus to find the slope, m, of the curve a
t this point and the equation " }{XPPEDIT 18 0 "y = m(x-2.6)+f(2.6);" 
"6#/%\"yG,&-%\"mG6#,&%\"xG\"\"\"$\"#E!\"\"!\"\"F+-%\"fG6#$\"#E!\"\"F+
" }{TEXT -1 1 " " }{TEXT 261 3 "to " }{TEXT -1 0 "" }{TEXT 260 14 "plo
t the line." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "df:=x->D(f)(
x):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "x0:=2.600:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "m:=df(x0):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 10 "y0:=f(x0):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 19 "TL:=m*(x-26/10)+y0:" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 48 "p2:=plot(TL,x=0.5..4.5,color=green,thickness=3):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(p1,p2);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 258 96 "We create an animation to
 show \"what the calculus is doing\" when finding the slope of the cur
ve." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "X:=[4,3.8,3.6,3.4,3.
2,3,2.8,2.7,2.65,2.601]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 
"ms:=x1->(f(x1)-y0)/(x1-x0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 27 "L:=x1->ms(x1)*(x-x1)+f(x1):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 50 "pL:=x1->plot(L(x1),x=1..5,color=blue,thickness=2):" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 124 "T:=textplot(\{[2.5,40,cat
(\"m = \",convert(evalf(m,5),string)),align=\{ABOVE,RIGHT\}]\},font=[T
IMES,BOLD,14],align=\{ABOVE,RIGHT\}):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 185 "S:=x1->textplot(\{[2.5,35,cat(\"ms = \",convert(eval
f(ms(x1),5),string))],[2.5,30,cat(\"x1 - x0 =\",convert(evalf(x1-x0,5)
,string))],[x1,-14,\"x1\"]\},font=[TIMES,BOLD,14],align=\{ABOVE,RIGHT
\}):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "M0:=textplot([2.6,-
10,\"xo\"],font=[TIMES,BOLD,14]):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 45 "vL:=x1->line([x0,y0],[x0,f(x1)],color=black):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "hL:=x1->line([x0,f(x1)],[x1,
f(x1)],color=red,thickness=5):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 46 "VL0:=x1->line([x0,-8],[x0,f(x1)-2],color=red):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "VL1:=x1->line([x1,-12],[x1,f(x1)-2]
,color=blue):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "B:=\{T,M0
\}:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 73 "fms:=seq(display(p2,
p1,VL0(x1),VL1(x1),hL(x1),vL(x1),S(x1),pL(x1)),x1=X):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 34 "pic:=display(fms,insequence=true):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(B,pic);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "expand(x*(4-x)*(x+2));" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 12 " INTEGRATION" }}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 20 "Sw
eeping Out an Area" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 199 "     The ap
proximation of the \"area under a curve\" by a suitable Riemann sum im
proves as the norm of the partition is made smaller.  Maple provides a
 way of illustrating this through commands in the " }{TEXT 287 7 "stud
ent" }{TEXT -1 99 " package.  These commands are easy to use and, usin
g the uniform partition, are suggestively named " }{TEXT 288 32 "leftb
ox, rightbox, and middlebox" }{TEXT -1 30 ".  The corresponding sums a
re " }{TEXT 289 18 "leftsum, rightsum," }{TEXT -1 5 " and " }{TEXT 
290 9 "middlesum" }{TEXT -1 14 ".  The use of " }{TEXT 291 8 "leftbox \+
" }{TEXT -1 4 "and " }{TEXT 292 8 "leftsum " }{TEXT -1 86 "is illustra
ted.  As the procedure is typical, the use of the others will be omitt
ed.  " }}{PARA 0 "" 0 "" {TEXT -1 620 "     Fairly early in a first ca
lculus course, however, it becomes desirable to introduce, particularl
y to engineering students, the idea of \"generating\" the area by \"sw
eeping it out\" with a vertical line segment.  Of course, this is only
 a gloss on Liebniz's error of passing to the limit before summing and
 so it makes many of us uneasy to teach it, but, faced with a multivar
iable calculus class of normal abilities, our principles may weaken an
d we may find ouselves sweeping out volumes just as we did when we wer
e students.  The single variable case will be illustrated second and h
igher 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 cl
assic case: the area \"under the curve\" for f(x) = " }{XPPEDIT 18 0 "
x(3-x);" "6#-%\"xG6#,&\"\"$\"\"\"F$!\"\"" }{TEXT -1 24 " on the interv
al [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 rectang
les.  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 -1 583 "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 completed \+
as the geometrical equivalent of the partial sum to this point.  We ca
n make a Maple animation of this using " }{TEXT 262 7 "leftbox" }
{TEXT -1 2 " ." }}}{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,color=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 0 105 "
T:=textplot(\{[1,1.3,\"Area = 4/3\"],[1,1.1,cat(\" Sum = \",convert(S(
n),string))]\},font=[HELVETICA,BOLD,14]):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 15 "display(p,R,T);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 109 "Tn:=i->textplot(\{[1,1.3,\"Area = 4/3\"],[1,1.1,cat(
\" Sum = \",convert(S(i),string))]\},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,in
sequence=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "One can genera
te the sweep without using " }{TEXT 263 7 "leftbox" }{TEXT -1 93 ".  H
ere's how.  Note that the X function which generates the partition poi
nts is still used. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "r1:=
k->rectangle([X(k),f(X(k))],[X(k+1),0],color=green):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 47 "s3:=seq(display(p,seq(r1(k),k=0..m)),m=0.
.100):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "display(s3,insequ
ence=true);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 230 "Now, one would li
ke to do away with 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 from 0 to the current positon of the red line is display
ed." }}}{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),thickness=4):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "pi
c:=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 = \",conver
t(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 1 {PARA 4 "" 0 "" {TEXT -1 21 "Volumes of \+
Revolution" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:with(p
lots):with(plottools):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 264 11 "Example 3: " }
{TEXT -1 4 "y = " }{XPPEDIT 18 0 "sqrt(x);" "6#-%%sqrtG6#%\"xG" }
{TEXT -1 39 "  on [0, 4].  (A classic illustration.)" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 120 "b2:=animate3d([x,sqrt(x)*cos(k*t),sqrt(x
)*sin(k*t)],x=0..4,k=0..2*Pi,t=0..1,view=[-1..5,-2..2,-2..2],color=k,f
rames=30):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "display(b2,insequence
=true,axes=normal,scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 265 
9 "Example 4" }{TEXT -1 6 "  x = " }{XPPEDIT 18 0 "y^2;" "6#*$%\"yG\"
\"#" }{TEXT -1 41 " on [0, 2]. (The companion of Example 3.)" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 112 "b3:=animate3d([y^2*cos(k*t)
,y,y^2*sin(k*t)],y=0..2,k=0..2*Pi,t=0..1,view=[-4..4,-1..3,-4..4],colo
r=k,frames=30):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 60 "display(
b3,insequence=true,axes=normal,scaling=constrained);" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 47 "display(b2,b3,axes=normal,scaling=constra
ined);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 10 "Vol(Disks)" }}{EXCHG {PARA 4 "" 0 "" 
{TEXT -1 30 "Volumes of Rotation by Slicing" }}}{EXCHG {PARA 5 "" 0 "
" {TEXT -1 34 "In Two Dimensions: Blackboard View" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 18 "Suppose we rotate " }{XPPEDIT 18 0 "y = sqrt(x);" 
"6#/%\"yG-%%sqrtG6#%\"xG" }{TEXT -1 62 " about the x-axis and look at \+
the approximating disks edge-on." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 34 "restart:with(student):with(plots):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 6 "m:=10;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 66 "b1:=leftbox(sqrt(x),x=0..4,m,shading=green,color=red,
thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "b2:=leftbo
x(-sqrt(x),x=0..4,m,shading=green,color=red,thickness=3):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "display(b1,b2);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 19 "In \+
Three Dimensions" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart
:with(plots):with(plottools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 192 
"We use parametric surfaces to draw in three dimensions a cylinder wit
h two ends .  We will use this as a \"slice\" to illustrate the method
 slicing for finding the volume of a solid of rotation." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 98 "cyl:=plot3d([x,cos(y),sin(y)],x=0..
1,y=0..2*Pi,scaling=constrained,color=green,style=patchnogrid):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "cyl2:=plot3d([0,r*cos(t),r*
sin(t)],r=0..1,t=0..2*Pi,scaling=constrained,color=red,style=patchnogr
id):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 101 "cyl3:=plot3d([1,r*
cos(t),r*sin(t)],r=0..1,t=0..2*Pi,scaling=constrained,color=red,style=
patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 35 "display(cy
l3,cyl2,cyl,axes=normal);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 238 "Cho
ose a function, f, and define it as a Maple function.  Then choose an \+
interval [a,b] over which to plot it by choosing a and b.  Finally, pl
ot it and turn it around to have a good look at it.  As an example, le
t us again choose f(x) = " }{XPPEDIT 18 0 "sqrt(x)" "6#-%%sqrtG6#%\"xG
" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "f:=x->s
qrt(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "a:=0;b:=4;" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "vol:=plot3d([x,f(x)*cos(t),f
(x)*sin(t)],x=a..b,t=0..2*Pi,color=yellow):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 67 "display(vol,axes=normal,orientation=[-115,75],scal
ing=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 53 "First, we de
fine the surface of the volume using the " }{TEXT 266 9 "wireframe" }
{TEXT -1 123 " style.  We will view the slices relative to this surfac
e to see how well the collection of slices approximates the volume." }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "surf:=plot3d([x,f(x)*cos(
t),f(x)*sin(t)],x=a..b,t=0..2*Pi,style=wireframe,color=black,grid=[10,
25]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 292 "To show the approximati
on of the volume by \"slicing\", we make a sequence of n slices, each \+
of thickness \"delta\", from x=a to x=b, where delta=(b-a)/n, and of r
adius r = f(X).  Here X can be chosen to be the x-coordinate of the le
ft-hand end of the slice, the right-hand end, the middle, etc. " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 6 "n:=20;" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 15 "delta:=(b-a)/n;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 27 "X:=[seq(a+j*delta,j=0..n)];" }}}{EXCHG {PARA 5 "" 0 "
" {TEXT -1 63 "1. Using the left-hand end of the slice to determine it
s radius" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "CYL:=k->plot3d
([x+X[k],f(X[k])*cos(y),f(X[k])*sin(y)],x=0..delta,y=0..2*Pi,color=gre
en,style=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "C
YL2:=k->plot3d([X[k],r*cos(t),r*sin(t)],r=0..f(X[k])+0.001,t=0..2*Pi,c
olor=red,style=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 101 "CYL3:=k->plot3d([X[k+1],r*cos(t),r*sin(t)],r=0..f(X[k])+0.001,t
=0..2*Pi,color=red,style=patchnogrid):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 1 " " }{TEXT 269 5 "Note:" }{TEXT -1 122 "   Here the plottin
g interval for \"r\" in CYL2 and CYL3 is artifically increased by 0.00
1 because of difficulties with the " }{TEXT 268 0 "" }{TEXT 267 6 "plo
t3d" }{TEXT -1 77 " command, which considers a plotting interval of le
ngth zero to be an error. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
49 "pic:=seq(display(CYL(k),CYL2(k),CYL3(k)),k=1..n):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "display(surf,pic,axes=normal,orient
ation=[-115,75],scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 30 "Now view this \"from the side\" " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 71 "display(surf,pic,axes=normal,orientation=[-90,90
],scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }
{TEXT 270 8 "Remark: " }{TEXT -1 111 "For functions which increase fas
ter than f(x) = x, the best pictures will be obtained by removing the \+
optional " }{TEXT 271 21 "scaling = constrained" }{TEXT -1 18 " comman
d from all " }{TEXT 272 7 "display" }{TEXT -1 10 " commands." }}}
{EXCHG {PARA 5 "" 0 "" {TEXT -1 40 "2. Using the right-hand end of the
 slice" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 108 "CYL:=k->plot3d([
x+X[k-1],f(X[k])*cos(y),f(X[k])*sin(y)],x=0..delta,y=0..2*Pi,color=gre
en,style=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 95 "C
YL2:=k->plot3d([X[k-1],r*cos(t),r*sin(t)],r=0..f(X[k]),t=0..2*Pi,color
=red,style=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 
"CYL3:=k->plot3d([X[k],r*cos(t),r*sin(t)],r=0..f(X[k]),t=0..2*Pi,color
=red,style=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 
"pic:=seq(display(CYL(k),CYL2(k),CYL3(k)),k=2..n+1):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 8 "display(" }{TEXT -1 0 "" }{MPLTEXT 1 0 64 
"surf,pic,axes=normal,orientation=[-115,75],scaling=constrained);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "display(surf,pic,axes=normal
,orientation=[-90,90],scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 66 "This has the obvious disadvantage that the surface cann
ot be seen." }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 22 "3. Using the mid-p
oint" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 122 "CYL:=k->plot3d([x+
X[k],f(X[k]+delta/2)*cos(y),f(X[k]+delta/2)*sin(y)],x=0..delta,y=0..2*
Pi,color=green,style=patchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 101 "CYL2:=k->plot3d([X[k],r*cos(t),r*sin(t)],r=0..f(X[k]
+delta/2),t=0..2*Pi,color=red,style=patchnogrid):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 103 "CYL3:=k->plot3d([X[k+1],r*cos(t),r*sin(t)],
r=0..f(X[k]+delta/2),t=0..2*Pi,color=red,style=patchnogrid):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "pic:=seq(display(CYL(k),CYL2
(k),CYL3(k)),k=1..n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "di
splay(surf" }{TEXT -1 0 "" }{MPLTEXT 1 0 60 ",pic,axes=normal,orientat
ion=[-115,75],scaling=constrained);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 71 "display(surf,pic,axes=normal,orientation=[-90,90],sca
ling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }{TEXT 
274 7 "Remark:" }{TEXT -1 29 "  This seems the best choice." }}}
{EXCHG {PARA 5 "" 0 "" {TEXT -1 26 "4.  Calculating the Volume" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "The approximate volume, using N di
sks and the mid-point approximation is given by" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 49 "A_vol:=Sum(Pi*(f(a+j*(b-a)/N))^2*(b-a)/N,j=0..
N);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "For any choice of N =N1 >
 0, its value may be found by making a choice of  N1 in the following \+
limit: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "L:=limit(value(A
_vol),N=N1);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "Thus the correct \+
volume is given by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "vol:=
limit(L,N1=infinity);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 43 "This agr
ees with the corresponding integral" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "I_vol:=Int(Pi*f(x)^2,x=a..b);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 19 "which has the value" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 13 "value(I_vol);" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 0 "
" }{TEXT 273 46 "5. Rotating the curve about the vertical axis:" }
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 67 "First, express th
e function as a function of z. Above, we used z = " }{XPPEDIT 18 0 "sq
rt(x);" "6#-%%sqrtG6#%\"xG" }{TEXT -1 23 " , so here we want x = " }
{XPPEDIT 18 0 "z^2;" "6#*$%\"zG\"\"#" }{TEXT -1 53 ".  Again, we use a
 Maple function, not an expression." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "g:=z->z^2;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 55 "Ne
xt, choose the interval of integration on the z-axis." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "c:=0;d:=2;" }}}{EXCHG {PARA 0 "" 0 
"" {TEXT -1 145 "Then choose the number of sub-intervals to be used.  \+
To begin, choose a small number, say, 5, then try again with 10 and fi
nally with perhaps 40." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "m:
=5;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 40 "This gives the length of a
 sub-interval." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "delta_z:=
(d-c)/m;" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 41 "This is the sequence \+
of partition points." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "Z:=
seq(c+i*delta_z,i=0..m);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 131 "We d
raw the surface of the object.  This will be used as a background to q
ualitatively estimate the accuracy to each approximation." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "vol2:=plot3d([g(z)*cos(t),g(z)*sin(
t),z],z=c..d,t=0..2*Pi,style=wireframe,color=black):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 48 "display(vol2,axes=normal,orientation=[-11
5,75]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "We define the lateral \+
surface of a slice and color it green.  The mid-point approximation is
 used." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 129 "CYLz:=k->plot3d(
[g(Z[k]+delta_z/2)*cos(y),g(Z[k]+delta_z/2)*sin(y),z+Z[k]],z=0..delta_
z,y=0..2*Pi,color=green,style=patchnogrid):" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 51 "We define the ends of the slice and color them red." }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "CYLz2:=k->plot3d([r*cos(t)
,r*sin(t),Z[k]],r=0..g(Z[k]+delta_z/2),t=0..2*Pi,color=red,style=patch
nogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 106 "CYLz3:=k->plot
3d([r*cos(t),r*sin(t),Z[k+1]],r=0..g(Z[k]+delta_z/2),t=0..2*Pi,color=r
ed,style=patchnogrid):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "We crea
te the collection of slices." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 53 "pic2:=seq(display(CYLz(k),CYLz2(k),CYLz3(k)),k=1..m):" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 51 "We display them together with the \+
object's surface." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "display
(" }{TEXT -1 0 "" }{MPLTEXT 1 0 46 "vol2,pic2,axes=normal,orientation=
[-120,120]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "We view our pictu
re \"on edge\"." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "display(
vol2,pic2,axes=normal,orientation=[-120,90]);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 5 "" 0 "" {TEXT -1 26 "6.  Calc
ulating the Volume" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 81 "The approxi
mate volume, using M disks and the mid-point approximation is given by
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "A_vol2:=Sum(Pi*(g(c+i*(
d-c)/M))^2*(d-c)/M,i=0..M);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 102 "F
or any choice of M =M1 > 0, its value may be found by making a choice \+
of  M1 in the following limit: " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 30 "L2:=limit(value(A_vol2),M=M1);" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 35 "Thus the correct volume is given by" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 28 "vol2:=limit(L2,M1=infinity);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 43 "This agrees with the corresponding integr
al" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 30 "I_vol2:=Int(Pi*g(z)^2
,z=c..d);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 19 "which has the value
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "value(I_vol2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "
" {TEXT -1 11 "Vol(Shells)" }}{EXCHG {PARA 18 "" 0 "" {TEXT -1 41 "Vol
umes of Rotation by Cylindrical Shells" }}{PARA 19 "" 0 "" {TEXT -1 
13 "J. A. Ziegler" }}{PARA 19 "" 0 "" {TEXT -1 17 "February 24, 2001" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:with(plots):with(
plottools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 203 "We define a funct
ion, f, as a Maple function, choose an interval [a,b] on which to defi
ne it, the number of subintervals to use, the width of a subinterval, \+
and the sequence of partition points of [a,b]." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "f:=x->4*x-x^2-3;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "a:=1;b:=3;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
5 "n:=5;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "delta:=(b-a)/n;
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "p:=[seq(a+j*delta,j=0..
n)];" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 114 "We rotate the curve on [
a,b] about the z-axis.  Enlarge and rotate the picture.  Be sure to lo
ok at it from below." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "vol
:=plot3d([x*cos(t),x*sin(t),f(x)],x=a..b,t=0..2*Pi):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 77 "display(vol,axes=normal,orientation=[45,6
5],color=green,scaling=constrained);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 62 "We create the surface with which to compare our approximation.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "surf:=plot3d([x*cos(t),
x*sin(t),f(x)],x=a..b,t=0..2*Pi,style=wireframe,color=black,grid=[10,2
5]):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 33 "For convenience, we now c
hoose a " }{TEXT 276 5 "style" }{TEXT -1 38 " for the surfaces of the \+
shells and a " }{TEXT 277 4 "grid" }{TEXT -1 25 " over which to plot t
hem." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "S:=patchnogrid:" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "G:=[25,25]:" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 37 "We draw a single cylindrical shell.  " }
{TEXT 275 8 "Remark: " }{TEXT -1 120 "In order that the surface may be
 compared to the shells, the mid-point approximation is convenient, bo
th here and below." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 98 "We begin by
 choosing our single shell by choosing a value for k.  This, of course
, may be changed." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 5 "k:=2;" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 99 "cyl0:=plot3d([p[k]*cos(y)
,p[k]*sin(y),z],z=0..f(p[k]+delta/2)+0.001,y=0..2*Pi,color=green,style
=S):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 103 "cyl1:=plot3d([p[k+
1]*cos(y),p[k+1]*sin(y),z],z=0..f(p[k]+delta/2)+0.001,y=0..2*Pi,color=
green,style=S):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 79 "cyl2:=pl
ot3d([r*cos(t),r*sin(t),0],r=p[k]..p[k+1],t=0..2*Pi,color=red,style=S)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 93 "cyl3:=plot3d([r*cos(t)
,r*sin(t),f(p[k]+delta/2)],r=p[k]..p[k+1],t=0..2*Pi,color=red,style=S)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 86 "display(surf,cyl3,cyl2
,cyl0,cyl1,axes=normal,orientation=[45,60],scaling=constrained);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "We create the sequence of cylindri
cal shells, then the sequence of pictures, and display these." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 109 "CYL0:=k->plot3d([p[k]*cos(y
),p[k]*sin(y),z],z=0..f(p[k]+delta/2)+0.001,y=0..2*Pi,color=green,styl
e=S,grid=G):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 113 "CYL1:=k->p
lot3d([p[k+1]*cos(y),p[k+1]*sin(y),z],z=0..f(p[k]+delta/2)+0.001,y=0..
2*Pi,color=green,style=S,grid=G):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 89 "CYL2:=k->plot3d([r*cos(t),r*sin(t),0],r=p[k]..p[k+1],
t=0..2*Pi,color=red,style=S,grid=G):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 103 "CYL3:=k->plot3d([r*cos(t),r*sin(t),f(p[k]+delta/2)],
r=p[k]..p[k+1],t=0..2*Pi,color=red,style=S,grid=G):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 58 "pic:=seq(display(CYL0(k),CYL1(k),CYL2(k),
CYL3(k)),k=1..n):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "displa
y(surf,pic,axes=normal,orientation=[45,65],scaling=constrained);" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 283 "The picture should be enlarged, t
hen rotated.  Now go back and change n to a larger number.  When the s
ingle shell is drawn, a larger value of k may be chosen, too, but care
 should be taken to reset this to 2 before choosing a new function and
 beginning anew with a small value of n." }}}}}{SECT 1 {PARA 3 "" 0 "
" {TEXT -1 22 " MULTIVARIBLE CALCULUS" }}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 11 "MVChainRule" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 290 "Here \+
is a fairly typical textbook problem. It is intended to be simply calc
ulational and, as such, is rather boring.  However, if Maple is used t
o illustrate its geometric aspect, this is quite interesting and give \+
one the feeling that, after all, the calculation was rather a grand th
ing. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Consider the ellipses  \+
" }{XPPEDIT 18 0 "x^2/16+y^2/4 = 1;" "6#/,&*&%\"xG\"\"#\"#;!\"\"\"\"\"
*&%\"yG\"\"#\"\"%F)F*\"\"\"" }{TEXT -1 6 " and  " }{XPPEDIT 18 0 "x^2/
4+y^2/9 = 1;" "6#/,&*&%\"xG\"\"#\"\"%!\"\"\"\"\"*&%\"yG\"\"#\"\"*F)F*
\"\"\"" }{TEXT -1 48 ", and let the first be given parametrically via \+
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "x1:=4*cos(t):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "y1:=2*sin(t):" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 28 "while the second is given by" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 15 "x2:=2*sin(2*t):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 15 "y2:=3*cos(2*t):" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 92 "Then the distance between a point on the first ellipse an
d a point on the second is given by" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 29 "s:=sqrt((x2-x1)^2+(y2-y1)^2);" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 92 "Notice that Maple has carried out the substitutions \+
so that s is explicitly a function of t." }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 57 "The problem is to use the appropriate chain rule to find \+
" }{XPPEDIT 18 0 "ds/dt;" "6#*&%#dsG\"\"\"%#dtG!\"\"" }{TEXT -1 8 " at
 t = " }{XPPEDIT 18 0 "Pi;" "6#%#PiG" }{TEXT -1 120 ".  Maple, having \+
made the substitutions, simply finds the derivative directly and very \+
quickly.  Then the derivative is " }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "der_s:=diff(s,t);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 21 "And its value at t = " }{XPPEDIT 18 0 "Pi" "6#%#PiG" }{TEXT -1 
3 " is" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "simplify(eval(sub
s(t=Pi,der_s)),radical);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 302 "Howe
ver, a particularly interesting aspect of this problem is the complexi
ty of the geometry associated with it. The curves, being ellipses, are
 simple, but the behavior of the line (of length s !) joining two \"pl
anets\" moving on the \"orbits\" according to the given parameterizati
ons is certainly not. " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 63 "First, \+
we illustrate the geometric situation by drawing curves." }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "c1:=plot([x1,y1,t=0..2*Pi],color=re
d,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "c2:=plot
([x2,y2,t=0..2*Pi],color=blue,thickness=2):" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 35 "display(c1,c2,scaling=constrained);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 535 "Next, we create an animation showing poi
nts moving along the curves and a line segment joining them.  The rela
tive speeds of the points along the curves are as prescribed by the pa
rameterizations.  Note that while the red point makes one revolution, \+
the blue one makes two.  Experiment with the animation buttons.  Try r
educing the speed.  It is satisfying to reflect that we have learned h
ow to solve such a complicated problem as finding, as each instant, th
e rate of change of the length of the line segment joining the two pla
nets." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "X1:=4*cos(2*Pi*t/6
0):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Y1:=2*sin(2*Pi*t/60)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "X2:=2*sin(2*Pi*t/30):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Y2:=3*cos(2*Pi*t/30):" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "L:=line([X1,Y1],[X2,Y2],c
olor=black,thickness=3):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 51 
"p1:=disk([X1,Y1],0.15,color=red,style=patchnogrid):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 52 "p2:=disk([X2,Y2],0.15,color=blue,style=pa
tchnogrid):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "pic:=seq(dis
play(L,c1,c2,p1,p2),t=0..60):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 49 "display(pic,insequence=true,scaling=constrained);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 35 "A Volume in Rectangular Coordinates" }}{EXCHG {PARA 4 "" 0 "" 
{TEXT -1 23 "Sweeping Out the Volume" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 153 "A typical exercise in \+
evaluating double integrals by using iterated integrals is to 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 87 " and the x,y-plane, \+
over the region R enclosed by the curves x = 0, y = x + 3, and y = " }
{XPPEDIT 18 0 "x^2+1;" "6#,&*$%\"xG\"\"#\"\"\"\"\"\"F'" }{TEXT -1 51 "
.  Here R is \"y-simple.\"  The \"lower curve\" is y = " }{XPPEDIT 18 
0 "x^2+1;" "6#,&*$%\"xG\"\"#\"\"\"\"\"\"F'" }{TEXT -1 38 " and the \"u
pper curve\" is y = x + 3.  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "
The volume between the surface z = f(x,y) and the x,y-plane over the r
egion R is given by the integral " }{XPPEDIT 18 0 "Int(Int(z,x = R .. \+
``),y = `` .. ``);" "6#-%$IntG6$-F$6$%\"zG/%\"xG;%\"RG%!G/%\"yG;F-F-" 
}{TEXT -1 51 ".  Its value may be found via the iterated inegral " }
{XPPEDIT 18 0 "int(int(4-(x/2)^2-(y/3)^2,y = x^2+1 .. x+3),x = 0 .. 2)
;" "6#-%$intG6$-F$6$,(\"\"%\"\"\"*$*&%\"xGF*\"\"#!\"\"\"\"#F/*$*&%\"yG
F*\"\"$F/\"\"#F//F3;,&*$F-\"\"#F*\"\"\"F*,&F-F*\"\"$F*/F-;\"\"!\"\"#" 
}{TEXT -1 107 ".  Maple can be used in the following way to help us vi
sualize what is meant by \"sweeping out the volume\". " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "restart:with(plots):with(plottools)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "f:=(x,y)->4-(x/2)^2-(y
/3)^2:" }}}{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 80 "s:=plot3d(f(x,y),x=0..2,y=Lc(x)..Uc
(x),grid=[10,5],style=wireframe,color=black):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 49 "L1:=line([0,1,0],[0,3,0],color=blue,thickness=2)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "L2:=spacecurve(\{[x,Uc
(x),0],[x,Lc(x),0]\},x=0..2,color=blue,thickness=2):" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 11 "R:=\{L1,L2\}:" }}}{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 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 51 "SV:=seq(display(A(0),A(i/20+0.001),s,C,R),i=
0..40):" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 83 "display(SV,insequence=tr
ue,axes=normal,view=[0..2,0..5,0..4],orientation=[-23,56]);" }}}
{EXCHG {PARA 4 "" 0 "" {TEXT -1 20 "Generating the Solid" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 515 "A somewhat more elaborate procedure may \+
be used to \"generate the solid.\" This makes a good classroom demonst
ration, but is unnecessarily complicated for an introductory Maple lab
oratory. This procedure is a good illustration of the use of \"image-v
alued\" Maple functions.  (These are an invention of my own, although \+
very likely many others working with Maple have invented them, too.)  \+
The 0.001 in the functional arguments prevent the plotting commands fr
om producing what would be essentially \"empty plot\" errors." }}}
{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=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=patchnogrid):" }}}{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,thic
kness=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 0 159 "SV2:=seq(display(A(0),A(i/20+0.001),surf(i
/20+0.001),Lside(i/20+0.001),Uside(i/20+0.001),LLine(i/20+0.001),ULine
(i/20+0.001),Bline(i/20+0.001),s,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 1 {PARA 4 "" 0 "" {TEXT -1 33 "A Volume in
 Spherical Coordinates" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "re
start:with(plots):with(plottools):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
278 9 "Exercise:" }{TEXT -1 48 " Draw the volume enclosed below by the
 cone z = " }{XPPEDIT 18 0 "sqrt(x^2+y^2);" "6#-%%sqrtG6#,&*$%\"xG\"\"
#\"\"\"*$%\"yG\"\"#F*" }{TEXT -1 33 " and above by the hemisphere z = \+
" }{XPPEDIT 18 0 "sqrt(9-x^2-y^2);" "6#-%%sqrtG6#,(\"\"*\"\"\"*$%\"xG
\"\"#!\"\"*$%\"yG\"\"#F," }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 
0 "" }{TEXT 282 9 "Solution:" }{TEXT -1 158 " Note that when y = 0 and
 x > 0, the first equation reduces to z = x.  Likewise, when x = 0 and
 y > 0, z = y.  Thinking in spherical coordinates, we see that " }
{XPPEDIT 18 0 "phi = Pi/4;" "6#/%$phiG*&%#PiG\"\"\"\"\"%!\"\"" }{TEXT 
-1 116 " at the wall of the cone.  So the portion of the hemisphere we
 need may be drawn in the following way.  By choosing " }{TEXT 279 18 
"style = wireframe," }{TEXT -1 71 " we will be able to see into the so
lid.  It is convenient to use t for " }{XPPEDIT 18 0 "theta;" "6#%&the
taG" }{TEXT -1 11 " and p for " }{XPPEDIT 18 0 "phi;" "6#%$phiG" }
{TEXT -1 1 "," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 78 "S:=plot3d(
3,t=0..2*Pi,p=0..Pi/4,coords=spherical,style=wireframe,color=black):" 
}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "It is quite easy to draw the c
one as a parametric surface in spherical coordinates.    Along the wal
l of the cone, " }{XPPEDIT 18 0 "phi = Pi/4;" "6#/%$phiG*&%#PiG\"\"\"
\"\"%!\"\"" }{TEXT -1 10 "  , while " }{XPPEDIT 18 0 "rho;" "6#%$rhoG
" }{TEXT -1 22 " runs from 0 to 3 and " }{XPPEDIT 18 0 "theta;" "6#%&t
hetaG" }{TEXT -1 11 " from 0 to " }{XPPEDIT 18 0 "2*Pi;" "6#*&\"\"#\"
\"\"%#PiGF%" }{TEXT -1 50 ".  For ease of typing, we abbreviate \"thet
a\" by t." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 88 "c:=plot3d([rho
,t,Pi/4],rho=0..3,t=0..2*Pi,coords=spherical,style=wireframe,color=bla
ck):" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 23 "Now display the result." 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "display(S,c,scaling=const
rained,orientation=[-25,75]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 121 
"Even when fully magnified, there are just too many grid lines.  This \+
can be improved by inserting the optional command   " }{TEXT 280 1 " \+
" }{TEXT -1 72 "grid=[25,5]   in the definition of S and grid=[5,25] t
o that of c.  Thus" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 90 "S:=pl
ot3d(3,t=0..2*Pi,p=0..Pi/4,coords=spherical,style=wireframe,color=blac
k,grid=[25,5]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "c:=plot
3d([rho,t,Pi/4],rho=0..3,t=0..2*Pi,coords=spherical,style=wireframe,co
lor=black,grid=[5,25]):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "
display(S,c,scaling=constrained,orientation=[-25,75]);" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 20 "That is much better." }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 0 "" }{TEXT 283 69 "Finding the volume using iterated \+
integrals in spherical coordinates." }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
284 48 "Visualization of a possible order of integration" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 55 "Let us add a representative radius line t
o our picture." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 72 "R:=spacec
urve([r,0,Pi/6],r=0..3,coords=spherical,color=red,thickness=3):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "display(S,c,R,scaling=constr
ained,orientation=[-25,75]);" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 37 "I
t is clear that, inside our volume, " }{XPPEDIT 18 0 "rho;" "6#%$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#*&%#piG\"\"\"\"\"%!
\"\"" }{TEXT -1 20 " , independently of " }{XPPEDIT 18 0 "theta;" "6#%
&thetaG" }{TEXT -1 93 ".  Having realized this, we can now write the n
ecessary integrals.  Integrating in the order " }{XPPEDIT 18 0 "rho,ph
i,theta;" "6%%$rhoG%$phiG%&thetaG" }{TEXT -1 53 ", and remembering tha
t in spherical coordinates dV = " }{XPPEDIT 18 0 "rho^2;" "6#*$%$rhoG
\"\"#" }{XPPEDIT 18 0 "sin(phi);" "6#-%$sinG6#%$phiG" }{XPPEDIT 18 0 "
d*rho;" "6#*&%\"dG\"\"\"%$rhoGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d*ph
i;" "6#*&%\"dG\"\"\"%$phiGF%" }{TEXT -1 1 " " }{XPPEDIT 18 0 "d*theta;
" "6#*&%\"dG\"\"\"%&thetaGF%" }{TEXT -1 10 ", we have " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 68 "V:=Int(Int(Int(rho^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 285 
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#%$phiG" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "theta;" "6#%&thetaG" 
}{TEXT -1 84 " fixed, produces the line segment we have already seen. \+
 The second integration, of " }{XPPEDIT 18 0 "phi;" "6#%$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 th
is surface by using the following parametric surface, which we define \+
as a function of s in a way that will be useful in a moment.  We abbre
viate \"phi\" by p. " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "s1
:=s->plot3d([rho,2*Pi*s/60,p],rho=0..3,p=0..Pi/4,coords=spherical,colo
r=red,style=patchcontour):  " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 60 "display(S,c,s1(0),scaling=constrained,orientation=[-25,75]);" }}
}{EXCHG {PARA 0 "" 0 "" {TEXT -1 124 "Now we make use of our \"surface
-valued\" function, s1, to make an animation showing the surface \"swe
eping 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 sett
ing 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 "" {TEXT 
-1 0 "" }{TEXT 286 53 "A more elaborate illustration of the last integ
ration" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 203 "By redefining the top \+
and bottom surfaces as \"surface valued\" functions, we may make an an
imation that really generates the volume.  The arguments \"s + 0.001\"
 in the functions sa and ca are used to keep " }{TEXT 281 6 "plot3d" }
{TEXT -1 157 " from generating an error when s = 0.  The line, L, has \+
been added to improve visibility.  Again, set the maximum magnificatio
n before running the animation." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 89 "sa:=s->plot3d(3,t=0..2*Pi*s/60,p=0..Pi/4,coords=spherical,styl
e=patchnogrid,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=spherica
l,style=patchnogrid,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+0.001),ca(s+0.001)),s=0..60):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 71 "display(pic1,insequence=true,scaling=constrained,or
ientation=[-25,75]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}}}}{MARK "0 0 0" 0 }{VIEWOPTS 1 1 0 1 1 1803 }