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{SECT 0 {EXCHG {PARA 204 "" 0 "" {TEXT 243 8 "MATH 151" }}{PARA 204 ""
 0 "" {TEXT 202 29 "LABORATORY ASSIGNMENT Bonus 3" }}{PARA 204 "" 0 ""
 {TEXT 243 22 "GEOMETRIC APPLICATIONS" }}{PARA 204 "" 0 "" {TEXT 243 
14 "OF INTEGRATION" }}{PARA 204 "" 0 "" {TEXT 251 0 "" }}{PARA 208 "" 
0 "" {TEXT 252 0 "" }}{PARA 208 "" 0 "" {TEXT 228 265 "    The definit
e integral can be used to solve a variety of problems from geometry;  \+
examples are finding areas between curves, lengths of curves,  volumes
 and surface area of three dimensional solids, and centroids of plane \+
regions. In this laboratory we will use " }{TEXT 205 5 "MAPLE" }{TEXT 
228 138 " to calculate the area between curves and the volume of solid
s of revolution. The volume problem will also give us the opportunity \+
to use " }{TEXT 205 5 "MAPLE" }{TEXT 228 25 "'s 3D graphing abilities.
" }}{PARA 208 "" 0 "" {TEXT 252 0 "" }}{PARA 208 "" 0 "" {TEXT 206 19 
"AREA BETWEEN CURVES" }}{PARA 208 "" 0 "" {TEXT 228 38 "    In class w
e showed that if  f (x) " }{TEXT 206 1 ">" }{TEXT 228 160 " g (x) for \+
all x in [ a , b ] then the area  A  of the region enclosed by the cur
ves y = f (x),        y = g (x),  and the lines  x = a  and  x = b  is
 given by" }}{PARA 204 "" 0 "" {TEXT 221 4 "A = " }{XPPEDIT 18 0 "Int(
f(x)-g(x), x = a .. b);" "6#-%$IntG6$,&-%\"fG6#%\"xG\"\"\"-%\"gGF)!\"
\"/F*;%\"aG%\"bG" }{TEXT 221 1 " " }{TEXT 221 2 " ." }}{PARA 208 "" 0 
"" {TEXT 228 109 "    In practice, a large part of the problem in usin
g this formula involves determining exactly where  f (x) " }{TEXT 206 
1 ">" }{TEXT 228 23 " g (x) and where g (x) " }{TEXT 206 1 ">" }{TEXT 
228 38 " f (x). We illustrate with an example:" }}{PARA 208 "" 0 "" 
{TEXT 252 0 "" }}{PARA 208 "" 0 "" {TEXT 228 2 "  " }{TEXT 206 7 "EXAM
PLE" }}{PARA 208 "" 0 "" {TEXT 252 0 "" }}{PARA 208 "" 0 "" {TEXT 228 
93 "  Find the total area of all regions enclosed by the curves f(x) =
 x + sin(2x)  and   g(x) = " }{XPPEDIT 18 0 "x^3;" "6#*$)%\"xG\"\"$\"
\"\"" }{TEXT 228 1 " " }{TEXT 228 1 "." }}{PARA 208 "" 0 "" {TEXT 252 
0 "" }}{PARA 208 "" 0 "" {TEXT 228 46 "  The first thing to do is to p
lot the graphs:" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 235 12 "with
(plots):" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 235 17 "f:=x->x+sin
(2*x);" }}{PARA 203 "" 1 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 
235 10 "g:=x->x^3;" }}{PARA 203 "" 1 "" }}{EXCHG {PARA 208 "> " 0 "" 
{MPLTEXT 1 235 41 "c:=textplot(\{[1.3,4,`g`],[3,1.5,`f`]\}):" }}}
{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 235 31 "p:=plot(\{f,g\},-Pi..Pi,
-5..5):" }}}{EXCHG {PARA 208 "> " 0 "" {TEXT 252 0 "" }}{PARA 208 "> "
 0 "" {MPLTEXT 1 235 17 "display(\{c,p\});" }}{PARA 210 "" 1 "" 
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p@=Pr#F_al7$Fg`l$\"1EMtewi+JF_al-%&COLORG6)%$RGBG$\"#5F.$\"\"!F.FhjlFh
jlFfjlFhjl-%*LINESTYLEG6#\"\"\"-%%VIEWG6$;$!+aEfTJ!\"*$\"+aEfTJFd[m;$!
#]F.$\"#]F.-%*GRIDSTYLEG6#%,RECTANGULARG-%+PROJECTIONG6#Ffjl-Fcjl6#%%N
ONEG-%%FONTG6$%*HELVETICAGFgjl" 1 2 2 1 10 1 2 9 1 4 2 1.0 45.0 45.0 1
 0 "Curve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 208 "" 0 "" {TEXT 
228 161 "Note that the curves intersect at the origin, since  f(0) = g
 (0) = 0. Next, we try find the X-coordinates of the other two points \+
of intersection of the curves." }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT
 1 235 19 "solve(f(x)=g(x),x);" }}{PARA 203 "" 1 "" }}{EXCHG {PARA 208
 "" 0 "" {TEXT 228 132 "Apparently there is no \"nice\" formula for th
e solutions, but we can still use \"fsolve\" to find decimal represent
ations for them." }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 235 33 "X[1
]:=fsolve(f(x)=g(x),x=-2..-1);" }}{PARA 203 "" 1 "" }}{EXCHG {PARA 208
 "> " 0 "" {MPLTEXT 1 235 33 "X[2]:=fsolve(f(x)=g(x),x,x=1..2);" }}
{PARA 203 "" 1 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 228 55 "Now we dete
rmine the interval or intervals where  f(x) " }{TEXT 206 1 ">" }{TEXT 
228 50 " g(x), and the interval or intervals where   g(x) " }{TEXT 
206 1 ">" }{TEXT 228 45 " f(x).From the graph, one can see that g (x) \+
" }{TEXT 206 1 ">" }{TEXT 228 18 " f(x) for x in  [ " }{XPPEDIT 18 0 "
X[1];" "6#&%\"XG6#\"\"\"" }{TEXT 228 1 " " }{TEXT 228 21 ", 0 ] and th
at f (x) " }{TEXT 206 1 ">" }{TEXT 228 22 " g (x) for x in [ 0 , " }
{XPPEDIT 18 0 "X[2];" "6#&%\"XG6#\"\"#" }{TEXT 228 1 " " }{TEXT 228 3 
" ]." }}{PARA 208 "" 0 "" {TEXT 252 0 "" }}{PARA 208 "" 0 "" {TEXT 
228 51 "Finally, we set up the integrals and evaluate them:" }}{PARA 
208 "" 0 "" {TEXT 252 0 "" }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 
235 56 "Area:=Int(g(x)-f(x),x=X[1]..0)+Int(f(x)-g(x),x=0..X[2]);" }}
{PARA 203 "" 1 "" }}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 235 9 "value
(%);" }}{PARA 212 "" 1 "" {XPPMATH 20 "6#$\"+;s.X@!\"*" }{TEXT 238 1 "
 " }}}{EXCHG {PARA 208 "" 0 "" {TEXT 252 0 "" }}{PARA 208 "" 0 "" 
{TEXT 228 4 "    " }}{PARA 208 "" 0 "" {TEXT 228 5 "     " }}{PARA 208
 "" 0 "" {TEXT 252 0 "" }}{PARA 208 "" 0 "" {TEXT 252 0 "" }}{PARA 208
 "" 0 "" {TEXT 206 31 "VOLUMES OF SOLIDS OF REVOLUTION" }}{PARA 208 ""
 0 "" {TEXT 228 84 "    A solid generated by revolving a curve  y = f \+
(x)  about the X-axis is called a " }{TEXT 231 19 "solid of revolution
" }{TEXT 228 71 ". Examples of such solids are cylinders, cones and sp
heres. The command" }}{PARA 204 "" 0 "" {TEXT 221 57 "plot3d([x, f(x)*
cos(t), f(x)*sin(t)], x=a..b, t=0..2*Pi);" }}{PARA 208 "" 0 "" {TEXT 
228 57 "plots the surface generated by revolving  y = f (x) ,  a " }
{TEXT 206 1 "<" }{TEXT 228 3 " x " }{TEXT 206 1 "<" }{TEXT 228 50 " b \+
about the X-axis.  As an example, let  f (x) = " }{XPPEDIT 18 0 "x^2;"
 "6#*$)%\"xG\"\"#\"\"\"" }{TEXT 228 1 " " }{TEXT 228 5 " , 0 " }{TEXT 
206 1 "<" }{TEXT 228 3 " x " }{TEXT 206 1 "<" }{TEXT 228 3 " 1." }}}
{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 235 8 "restart:" }}}{EXCHG 
{PARA 208 "> " 0 "" {MPLTEXT 1 235 12 "with(plots):" }}}{EXCHG {PARA 
208 "> " 0 "" {MPLTEXT 1 235 10 "f:=x->x^2;" }}{PARA 213 "" 1 "" 
{XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(*$)F'\"\"#\"\"
\"F(F(F(" }{TEXT 238 1 " " }}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 
235 57 "plot3d([x, f(x)*cos(t), f(x)*sin(t)], x=0..1, t=0..2*Pi);" }}
{PARA 210 "" 1 "" {GLPLOT3D 641 487 487 {PLOTDATA 3 "6&-%*AXESSTYLEG6#
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000000000000-%*LINESTYLEG6#\"\"\"-%%FONTG6$%*HELVETICAG\"#5" 1 2 5 1 
10 1 2 1 1 1 2 1.0 45.0 45.0 1 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 
208 "" 0 "" {TEXT 228 82 "The formula for the volume V of the solid re
sulting from rotating  y = f (x) ,  a " }{TEXT 206 1 "<" }{TEXT 228 3 
" x " }{TEXT 206 1 "<" }{TEXT 228 24 " b , about the X-axis is" }}
{PARA 204 "" 0 "" {TEXT 221 2 "V=" }{XPPEDIT 18 0 "Pi*Int(f(x)^2, x = \+
a .. b);" "6#*&%#PiG\"\"\"-%$IntG6$*$)-%\"fG6#%\"xG\"\"#F%/F.;%\"aG%\"
bGF%" }{TEXT 221 1 " " }}{PARA 208 "" 0 "" {TEXT 228 26 "Thus, in the \+
above example" }}{PARA 204 "" 0 "" {TEXT 221 4 "V = " }{XPPEDIT 18 0 "
Pi;" "6#%#PiG" }{TEXT 221 1 " " }{XPPEDIT 18 0 "Int((x^2)^2, x = 0 .. \+
1);" "6#-%$IntG6$*$)*$)%\"xG\"\"#\"\"\"F+F,/F*;\"\"!F," }{TEXT 221 1 "
 " }{TEXT 221 1 "," }}{PARA 208 "" 0 "" {TEXT 228 13 "and therefore" }
}}{EXCHG {PARA 208 "> " 0 "" {MPLTEXT 1 235 22 "V:=int(Pi*x^4,x=0..1);
" }}{PARA 203 "" 1 "" }}{EXCHG {PARA 208 "" 0 "" {TEXT 228 4 "    " }
{TEXT 206 9 "EXERCISES" }}{PARA 208 "" 0 "" {TEXT 228 71 "     1. Find
 the  total area of all regions enclosed by the curves y = " }
{XPPEDIT 18 0 "x^2*cos(x);" "6#*&)%\"xG\"\"#\"\"\"-%$cosG6#F%F'" }
{TEXT 228 1 " " }{TEXT 228 10 " and  y = " }{XPPEDIT 18 0 "x^3-x;" "6#
,&*$)%\"xG\"\"$\"\"\"F(F&!\"\"" }{TEXT 228 1 " " }{TEXT 228 1 "." }}
{PARA 208 "" 0 "" {TEXT 228 66 "     2. Make an appropriate choice of \+
f (x) to generate graphs of:" }}{PARA 208 "" 0 "" {TEXT 228 34 "      \+
    a) a sphere of radius 5," }}{PARA 208 "" 0 "" {TEXT 228 49 "      \+
    b) a cylinder of height 6 and radius 3," }}{PARA 208 "" 0 "" {TEXT
 228 52 "          c) a cone with height 6 and base radius 3." }}}
{EXCHG {PARA 208 "" 0 "" {TEXT 228 12 "        Use " }{TEXT 205 6 "MAP
LE " }{TEXT 228 30 "and the formula for the volume" }{TEXT 205 1 " " }
{TEXT 228 24 "of a solid of revolution" }{TEXT 205 1 " " }{TEXT 228 
102 "to find the volumes of the above solids. Show that your results a
re consistent with the usual formulas" }}{PARA 204 "" 0 "" {TEXT 221 
18 "Volume(Sphere) =  " }{XPPEDIT 18 0 "4*Pi*r^3/3;" "6#**\"\"%\"\"\"%
#PiGF%)%\"rG\"\"$F%F)!\"\"" }{TEXT 221 1 " " }{TEXT 221 2 " ," }}
{PARA 204 "" 0 "" {TEXT 221 76 "                                      \+
                   Volume(Cylinder) = " }{XPPEDIT 18 0 "Pi*r^2*h;" "6#
*(%#PiG\"\"\"*$)%\"rG\"\"#F%F%%\"hGF%" }{TEXT 221 1 " " }{TEXT 221 
182 "                                                                 \+
                                                                      \+
                           Volume(Cone) = (1/3)" }{XPPEDIT 18 0 "Pi;" 
"6#%#PiG" }{TEXT 221 1 " " }{XPPEDIT 18 0 "r^2;" "6#*$)%\"rG\"\"#\"\"
\"" }{TEXT 221 1 " " }}{PARA 208 "" 0 "" {TEXT 252 0 "" }}{PARA 208 ""
 0 "" {TEXT 228 21 "      3. The command " }}{PARA 204 "" 0 "" {TEXT 
221 84 "animate3d([x, f(x)*cos(t*d), f(x)*sin(t*d)], x=a..b, t=0..2*Pi
,d=0..1, axes=NORMAL);" }}{PARA 208 "" 0 "" {TEXT 228 320 " produces a
 \"movie\" of the generation of the surface. Try it. (you don't have t
o turn it in.) To get a less jerky movie, add \"frames=30\" after \"d=
0..1\" in the command. (Don't go overboard with the number of frames--
Your computer may object to the memory demands of 3d animations. Save \+
your work before proceeding.)" }}}{PARA 209 "" 0 "" {TEXT 253 0 "" }}}

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