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{SECT 0 {EXCHG {PARA 257 "" 0 "" {TEXT -1 7 "MAT 156" }}{PARA 258 "" 
0 "" {TEXT -1 5 "Lab 7" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 
0 "" {TEXT -1 33 "Topic 1: Area between two curves." }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 170 "We have seen how to graph two functions simult
aneously. We have also seen how to solve equations. With these two pro
cesses, we can calculate the area between two curves. " }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 44 "Example: Find the area between the parabo
la " }{XPPEDIT 18 0 "y = 2*x^2;" "6#/%\"yG*&\"\"#\"\"\"*$%\"xGF&F'" }
{TEXT -1 14 " and the line " }{XPPEDIT 18 0 "y = -2*x+4;" "6#/%\"yG,&*
&\"\"#\"\"\"%\"xGF(!\"\"\"\"%F(" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 25 "f:=x->2*x^2;g:=x->-2*x+4;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&arrowGF(,$*&\"\"#\"\"
\")9$F.F/F/F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"gGf*6#%\"xG6
\"6$%)operatorG%&arrowGF(,&*&\"\"#\"\"\"9$F/!\"\"\"\"%F/F(F(F(" }}}
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-%%VIEWG6$;F(Fgz%(DEFAULTG" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 
45.000000 0 0 "Curve 1" "Curve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
83 "By looking at the graph it seems rather obvious that the two curve
s intersect when " }{XPPEDIT 18 0 "x = 1;" "6#/%\"xG\"\"\"" }{TEXT -1 
10 " and when " }{XPPEDIT 18 0 "x = -2;" "6#/%\"xG,$\"\"#!\"\"" }
{TEXT -1 74 ". We can verify that by setting the two equations to be e
qual and solving:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "fsolve
(f(x)=g(x), x);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$$!+++++?!\"*$\"++++
+5F%" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 148 "For later purposes it wi
ll be interesting to give names to the two solutions. We achieve this \+
by labeling the result of the previous command by sol:" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "sol:=fsolve(f(x)=g(x), x);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$solG6$$!+++++?!\"*$\"+++++5F(" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 58 "Now we can recover the two solutio
ns as sol[1] and sol[2]:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "
sol[1];" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$!+++++?!\"*" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 7 "sol[2];" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#$\"+++++5!\"*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 59 "T
o find the area between the two curves we use the formula " }{XPPEDIT 
19 1 "int(f(x)-g(x),x = a .. b);" "6#-%$intG6$,&-%\"fG6#%\"xG\"\"\"-%
\"gG6#F*!\"\"/F*;%\"aG%\"bG" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "wh
ere the " }{XPPEDIT 18 0 "f(x);" "6#-%\"fG6#%\"xG" }{TEXT -1 25 " is t
he top function and " }{XPPEDIT 18 0 "g(x);" "6#-%\"gG6#%\"xG" }{TEXT 
-1 30 " is the bottom function. Also " }{XPPEDIT 18 0 "a;" "6#%\"aG" }
{TEXT -1 8 " is the " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 44 "-coo
rdinate of the  point furthest left and " }{XPPEDIT 18 0 "b;" "6#%\"bG
" }{TEXT -1 8 " is the " }{XPPEDIT 18 0 "x;" "6#%\"xG" }{TEXT -1 92 "-
coordinate of the point furthest to the right on the region we study. \+
Since in our example " }{XPPEDIT 18 0 "f(x) < g(x);" "6#2-%\"fG6#%\"xG
-%\"gG6#F'" }{TEXT -1 39 " in the region we are interested we do:" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "A:=Int(g(x)-f(x), x=sol[1]..
sol[2]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%$IntG6$,(*&\"\"#\"
\"\"%\"xGF+!\"\"\"\"%F+*&F*F+)F,F*F+F-/F,;$!+++++?!\"*$\"+++++5F5" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(A);" }}{PARA 11 "" 1 "
" {XPPMATH 20 "6#$\"\"*\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 257 52 "Assignment Part 1:  Fin
d the area between the curve " }{XPPEDIT 258 0 "f(x) = x+1;" "6#/-%\"f
G6#%\"xG,&F'\"\"\"F)F)" }{TEXT 259 5 " and " }{XPPEDIT 260 0 "g(x) = x
^2;" "6#/-%\"gG6#%\"xG*$F'\"\"#" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT 265 57 "Assignment Part 2:  Find the area
 between the two curves " }{XPPEDIT 261 0 "y = x^6;" "6#/%\"yG*$%\"xG
\"\"'" }{TEXT 262 5 " and " }{XPPEDIT 263 0 "y = x+2;" "6#/%\"yG,&%\"x
G\"\"\"\"\"#F'" }{TEXT -1 1 "." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 11 "" 1 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT 264 31 "Topic 2: Volumes of revolution." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 217 "
Example:  Let R be the arc bounded above by the graph of f(x) = x^2-1 \+
and below by the x-axis.   Compute the volume of the solid obtained by
 rotating R about the x-axis.  We can use the maple student calculus p
ackage " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "with(Student[Cal
culus1]): " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "f:=x->x^2-1;
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"fGf*6#%\"xG6\"6$%)operatorG%&a
rrowGF(,&*$)9$\"\"#\"\"\"F1F1!\"\"F(F(F(" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 21 "sol:= fsolve(f(x)=0);" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#>%$solG6$$!+++++5!\"*$\"\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 79 "V:=VolumeOfRevolution(f(x), x=sol[1]..sol[2], output=
integral,axis=horizontal);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG-%
$IntG6$*&%#PiG\"\"\"),&*$)%\"xG\"\"#F*F*F*!\"\"F0F*/F/;$!+++++5!\"*$F*
\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "value(V);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#$\"+k@.^L!\"*" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 63 "Now compute the volume obtained by rotating about the lin
e x=-2" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 97 "V:=VolumeOfRevolu
tion(f(x), x=sol[1]..sol[2], output=integral,axis=vertical,distancefro
maxis=-2);" }{TEXT -1 0 "" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"VG-%$
IntG6$,$**\"\"#\"\"\"%#PiGF+,&%\"xGF+F*F+F+,&*$)F.F*F+F+F+!\"\"F+F2/F.
;$!+++++5!\"*$F+\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "val
ue(V);" }{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 103 "To view the sol
id we can use Maple's plot3d command:  the \"t\" parameter measures th
e angle of rotation:" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#$\"+#3;bn\"!\"
)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 70 "plot3d([x, f(x)*cos(t)
, f(x)*sin(t)], x=-1..1, t=0..2*Pi, axes=BOXED);" }}{PARA 13 "" 1 "" 
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urve 1" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 266 18 "Assignment Part 3:" }
{TEXT -1 52 " Let R be the region bounded  above by the graph of " }
{XPPEDIT -1 0 "f(x) = sqrt(4-x^2);" "6#/-%\"fG6#%\"xG-%%sqrtG6#,&\"\"%
\"\"\"*$F'\"\"#!\"\"" }{TEXT -1 26 " and below by the x-axis. " }}
{PARA 0 "" 0 "" {TEXT 267 3 " A)" }{TEXT -1 75 "    Find the volume of
 the solid obtained by rotating R about the line x=-3" }}{PARA 0 "" 0 
"" {TEXT 268 2 "B)" }{TEXT -1 73 "    Find the volume of the solid obt
ained by rotating R about the y-axis." }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT 269 20 "Assignment Part 4:  " }{TEXT -1 81 "Find the volume of t
he ellipsoid obtained by rotating the area inside the ellipse" }}
{PARA 0 "" 0 "" {XPPEDIT 256 0 "x^2/9+y^2/49 = 1;" "6#/,&*&%\"xG\"\"#
\"\"*!\"\"\"\"\"*&%\"yGF'\"#\\F)F*F*" }{TEXT -1 0 "" }}{PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "about the x-axis." }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 284 "The method
s VolumeOfRevolution and plot3d  can be used to display the three dime
nsional images.  You can learn more about this by either reading the M
aple Help files  (accessible by typing ?plot3d   or ?VolumeOfRevolutio
n) or by looking at lab 7 on the department home page for 156.  " }}}}
{MARK "42 0 0" 116 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 
1 2 33 1 1 }