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{SECT 0 {PARA 3 "" 0 "" {TEXT -1 20 "Solids of Revolution" }{MPLTEXT 
1 0 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 55 "Generating a Solid by Rotating a \+
Plane Curve about the " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 6 "-axi
s " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "with(plots):\nwith(plo
ttools):" }}}{PARA 0 "" 0 "" {TEXT -1 72 "The following procedure will
 create an animation of a curve of the form " }{XPPEDIT 18 0 "y=f(x)" 
"6#/%\"yG-%\"fG6#%\"xG" }{TEXT -1 3 " , " }{XPPEDIT 18 0 "x" "6#%\"xG
" }{TEXT -1 5 " in [" }{XPPEDIT 18 0 "a,b" "6$%\"aG%\"bG" }{TEXT -1 
27 "], being rotated about the " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT 
-1 7 "-axis. " }}{PARA 0 "" 0 "" {TEXT -1 18 "The parameters are" }}
{PARA 16 "" 0 "" {XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 48 ", the funct
ion defining the curve to be rotated," }}{PARA 16 "" 0 "" {XPPEDIT 18 
0 "a" "6#%\"aG" }{TEXT -1 2 ", " }{XPPEDIT 18 0 "b" "6#%\"bG" }{TEXT 
-1 22 ", the interval on the " }{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 
41 "-axis over which to create the animation," }}{PARA 16 "" 0 "" 
{XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 46 ", the number of frames used \+
in the animation. " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 938 "Rotat
eX := proc(f, a, b, n)\nlocal k, theta, g, R, c, L, A, B, arc, T;\n   \+
for k from 1 to n do\n      theta := k*2*Pi/n:\n      g[k] := plot3d([
x, f(x)*cos(t), f(x)*sin(t)], x=a..b, t=0..theta):\n      R[k] := spac
ecurve([b, f(b)*cos(theta)*t, f(b)*sin(theta)*t], t=0..1, color=red):
\n      c[k] := spacecurve([b, 0.25*cos(t), 0.25*sin(t)], t=0..theta, \+
color=red):\n   od:\n   L := line([b,0,0],[b,1,0],color=red):\n   g[0]
 := spacecurve([x, f(x), 0], x=a..b, thickness=2):\n   R[0] := line([b
,f(b),0], [b,0,0], color=red):\n   c[0] := L:\n   A := display([seq(g[
k], k=0..n)], insequence=true, axes=normal, orientation=[-58,29], styl
e=patch):\n   B := display([seq(R[k], k=0..n)], insequence=true, axes=
normal, orientation=[-58,29]):\n   arc := display([seq(c[k], k=1..n)],
 insequence=true, axes=normal, orientation=[-58,29]):\n   T := textplo
t3d([b, 0.4*cos(Pi/8), 0.4*sin(Pi/8), `q`], font=[SYMBOL,10], color=re
d):\n   display([A, B, arc, T, L]);\nend:" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 38 "Here is an example using the function " }{XPPEDIT 18 0 "y
=1/x" "6#/%\"yG*&\"\"\"\"\"\"%\"xG!\"\"" }{TEXT -1 18 " on the interva
l [" }{XPPEDIT 18 0 "1,5" "6$\"\"\"\"\"&" }{TEXT -1 17 "] with 15 fram
es." }}{PARA 0 "" 0 "" {TEXT -1 42 "Click on the graph, and run the an
imation." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "n := 15:\nf := x -> 1/x
:\na := 1: b:=5:\nRotateX(f, a, b, n);" }}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 31 "An Animation of the Disk Method" }}{PARA 0 "" 0 "" {TEXT 
-1 48 "The following procedure creates an animation of " }{XPPEDIT 18 
0 "n" "6#%\"nG" }{TEXT -1 46 " disks being stacked to illustrate the f
ormula" }}{PARA 0 "" 0 "" {TEXT -1 34 "                               \+
   " }{XPPEDIT 18 0 "V = int(Pi*(f(y))^2, y=c..d)" "6#/%\"VG-%$intG6$*
&%#PiG\"\"\"*$-%\"fG6#%\"yG\"\"#F*/F/;%\"cG%\"dG" }{TEXT -1 1 "," }}
{PARA 0 "" 0 "" {TEXT -1 70 "which computes the volume of the solid ob
tained by rotating the curve " }{XPPEDIT 18 0 "x = f(y)" "6#/%\"xG-%\"
fG6#%\"yG" }{TEXT -1 11 " about the " }{XPPEDIT 18 0 "y" "6#%\"yG" }
{TEXT -1 6 "-axis." }}{PARA 0 "" 0 "" {TEXT -1 18 "The parameters are
" }}{PARA 16 "" 0 "" {XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 15 ", the f
unction," }}{PARA 16 "" 0 "" {XPPEDIT 18 0 "c, d" "6$%\"cG%\"dG" }
{TEXT -1 54 ", the interval over which the disks are being stacked," }
}{PARA 16 "" 0 "" {XPPEDIT 18 0 "n" "6#%\"nG" }{TEXT -1 61 ", the numb
er of disks displayed (keep this parameter modest)." }}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 335 "DiskY := proc(f, c, d, n)\nlocal h, k;\n  \+
 h:=(d-c)/(n-1):\n   for k from 1 to n do\n      disk.k := cylinder([0
,0,c+(k-1)*h], f(c+(k-1)*h), h):\n      p.k := plots[display]([seq(dis
k.j, j=1..k)], scaling=constrained, style=patchnogrid, axes=framed, or
ientation=[45,65]):\n   od:\n   plots[display]([seq(p.k, k=1..n)], ins
equence=true);\nend: " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 25 "Here is \+
an example using " }{XPPEDIT 18 0 "x=2/y" "6#/%\"xG*&\"\"#\"\"\"%\"yG!
\"\"" }{TEXT -1 5 " on [" }{XPPEDIT 18 0 "1,4" "6$\"\"\"\"\"%" }{TEXT 
-1 16 "] using 6 disks." }}{PARA 0 "" 0 "" {TEXT -1 41 "Click on the g
raph and run the animation." }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 51 "n :=
 6:\nc:=1: d:=4:\nf := y->2/y;\nDiskY(f, c, d, n);" }}}}{SECT 1 {PARA 
4 "" 0 "" {TEXT -1 13 "Assignment 13" }}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 5 "Ex.1:" }}{PARA 0 "" 0 "" {TEXT -1 22 "Consider the function " }
{XPPEDIT 18 0 "f(x) = arctan((x-1)/(x+1))" "6#/-%\"fG6#%\"xG-%'arctanG
6#*&,&F'\"\"\"\"\"\"!\"\"F-,&F'F-\"\"\"F-F/" }{TEXT -1 5 " for " }
{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 5 " in [" }{XPPEDIT 18 0 "-1/2,4
" "6$,$*&\"\"\"\"\"\"\"\"#!\"\"F(\"\"%" }{TEXT -1 2 "]." }}{PARA 0 "" 
0 "" {TEXT -1 7 "a) Use " }{TEXT 256 7 "RotateX" }{TEXT -1 85 " to cre
ate an animation of the solid of revolution obtained by rotating the g
raph of " }{XPPEDIT 18 0 "f" "6#%\"fG" }{TEXT -1 12 " around the " }
{XPPEDIT 18 0 "x" "6#%\"xG" }{TEXT -1 6 "-axis." }}{PARA 0 "" 0 "" 
{TEXT -1 21 "b) Use the procedure " }{TEXT 256 5 "DiskY" }{TEXT -1 63 
" to create an illustration of the disk method for the function " }
{XPPEDIT 18 0 "f=f(y)" "6#/%\"fG-F$6#%\"yG" }{TEXT -1 7 ", with " }
{XPPEDIT 18 0 "y" "6#%\"yG" }{TEXT -1 5 " in [" }{XPPEDIT 18 0 "-1/2,4
" "6$,$*&\"\"\"\"\"\"\"\"#!\"\"F(\"\"%" }{TEXT -1 3 "]. " }}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 5 "Ex.2:" }}{PARA 0 "" 0 "" {TEXT -1 89 "An e
mbankment is to be built around a circular wading pool as shown in the
 figure below. " }}{PARA 0 "" 0 "" {TEXT -1 31 "The dimensions are as \+
follows: " }}{PARA 16 "" 0 "" {TEXT -1 39 "the inner radius (of the bl
ue area) is " }{XPPEDIT 18 0 "5*m" "6#*&\"\"&\"\"\"%\"mGF%" }{TEXT -1 
1 "," }}{PARA 16 "" 0 "" {TEXT -1 48 "the width of the embankment (the
 brown area) is " }{XPPEDIT 18 0 "6*m" "6#*&\"\"'\"\"\"%\"mGF%" }
{TEXT -1 1 "." }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 271 "restart;\n
with(plots):\np1:=plot3d([x*cos(t), x*sin(t), (11-x)*(x-5)^2/16], t=0.
.2*Pi, x=5..11, orientation=[45,65], color=brown, style=wireframe, sca
ling=constrained):\np2:=plot3d([x*cos(t), x*sin(t), 0], t=0..2*Pi, x=0
..5, color=blue, style=patchnogrid):\ndisplay([p1,p2]);" }}}{PARA 0 "
" 0 "" {TEXT -1 52 "The next figure shows the profile of the embankmen
t." }}{PARA 0 "" 0 "" {TEXT -1 6 "It is " }{XPPEDIT 18 0 "6*m" "6#*&\"
\"'\"\"\"%\"mGF%" }{TEXT -1 11 " wide, and " }{XPPEDIT 18 0 "2*m" "6#*
&\"\"#\"\"\"%\"mGF%" }{TEXT -1 20 " high at a distance " }{XPPEDIT 18 
0 "2*m" "6#*&\"\"#\"\"\"%\"mGF%" }{TEXT -1 21 " from the outer edge." 
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "plot((11-x)*(x-5)^2/16, x=
5..11, color=brown, scaling=constrained);" }}}{PARA 0 "" 0 "" {TEXT 
-1 27 "a) Find a cubic polynomial " }{XPPEDIT 18 0 "p" "6#%\"pG" }
{TEXT -1 30 " which fits the three points (" }{XPPEDIT 18 0 "5,0" "6$
\"\"&\"\"!" }{TEXT -1 4 "), (" }{XPPEDIT 18 0 "9,2" "6$\"\"*\"\"#" }
{TEXT -1 8 "), and (" }{XPPEDIT 18 0 "11,0" "6$\"#6\"\"!" }{TEXT -1 
57 "), and has a gentle slope at the inside, i.e., satisfies " }
{XPPEDIT 18 0 "D(p)(5)=0" "6#/--%\"DG6#%\"pG6#\"\"&\"\"!" }{TEXT -1 1 
"." }}{PARA 0 "" 0 "" {TEXT -1 13 "Hint: Set up " }{XPPEDIT 18 0 "p" "
6#%\"pG" }{TEXT -1 90 " as a general cubic polynomial, and determine i
ts coefficients by solving the 4 equations " }{XPPEDIT 18 0 "p(x[i])=y
[i]" "6#/-%\"pG6#&%\"xG6#%\"iG&%\"yG6#F*" }{TEXT -1 2 ", " }{XPPEDIT 
18 0 "i=1,2" "6$/%\"iG\"\"\"\"\"#" }{TEXT -1 9 ", 3, and " }{XPPEDIT 
18 0 "D(p)(5) = 0;" "6#/--%\"DG6#%\"pG6#\"\"&\"\"!" }{TEXT -1 1 "." }}
{PARA 0 "" 0 "" {TEXT -1 65 "b) Determine the amount of fill required \+
to build the embankment." }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 5 "Ex.3:
" }}{PARA 0 "" 0 "" {TEXT -1 21 " Consider the curves " }{XPPEDIT 18 
0 "y[1] = x/(3*x^2+1);" "6#/&%\"yG6#\"\"\"*&%\"xG\"\"\",&*&\"\"$F**$F)
\"\"#F*F*\"\"\"F*!\"\"" }{TEXT -1 5 " and " }{XPPEDIT 18 0 "y[2] = m*x
;" "6#/&%\"yG6#\"\"#*&%\"mG\"\"\"%\"xGF*" }{TEXT -1 1 "." }}{PARA 0 "
" 0 "" {TEXT -1 22 "a) For what values of " }{XPPEDIT 18 0 "m" "6#%\"m
G" }{TEXT -1 4 " do " }{XPPEDIT 18 0 "y[1]" "6#&%\"yG6#\"\"\"" }{TEXT 
-1 5 " and " }{XPPEDIT 18 0 "y[2]" "6#&%\"yG6#\"\"#" }{TEXT -1 15 " bo
und an area?" }}{PARA 0 "" 0 "" {TEXT -1 60 "b) Create a plot of the c
urves for some admissible value of " }{XPPEDIT 18 0 "m" "6#%\"mG" }
{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 44 "c) Find the enclosed are
a (as a function of " }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 2 ")." }}
{PARA 0 "" 0 "" {TEXT -1 81 "d) Create a plot that shows the dependenc
e of the area of the enclosed region on " }{XPPEDIT 18 0 "m" "6#%\"mG
" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 20 "Is there a value of \+
" }{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 41 " for which this area is m
aximal? Comment." }}{PARA 0 "" 0 "" {TEXT -1 14 "e) How should " }
{XPPEDIT 18 0 "m" "6#%\"mG" }{TEXT -1 51 " be chosen such that the enc
losed area equals 1000?" }}}}}{MARK "5" 0 }{VIEWOPTS 1 1 0 1 1 1803 }