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{SECT 0 {PARA 256 "" 0 "" {TEXT -1 13 "Assingment 2 " }{TEXT 257 9 "So
lutions" }}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 23 "Q.1. Consider the curv
e" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "x(t):= t*sin(t); y(t):=
t*cos(t); 0<=t, t<=Pi;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"xG6#%\"
tG*&F'\"\"\"-%$sinGF&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>-%\"yG6#%
\"tG*&F'\"\"\"-%$cosGF&F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$1\"\"!%\"
tG1F%%#PiG" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 18 "a) Graph the curve
" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 16 "Solution/Answwer" }}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "plot([x(t),y(t), t=0..Pi]);" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "inital_point, P1 = (0,0); end_point
, P2 = (0,-Pi);" }}{PARA 13 "" 1 "" {GLPLOT2D 157 301 301 {PLOTDATA 2 
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w!e8gT=\"F_q$!37V?h%ej&4CF_q7$$\"3f$4YRaNw/\"F_q$!3^C/J=`fVDF_q7$$\"3
\"*yACf@G5\"*F6$!3:2$Ql[C*fEF_q7$$\"3*y=8k94yT(F6$!3>w7&yS!3%y#F_q7$$
\"3+_W$ziI'=mF6$!3q=PbH97OGF_q7$$\"3)o#)y`s!=%z&F6$!3HNNjve&e)GF_q7$$
\"3;#=j\\Uk!))[F6$!3#[*4@f>AOHF_q7$$\"3,6RY<#[[&RF6$!3;\\)e$Gpr$)HF_q7
$$\"3roBZ-v5PIF6$!3Y$3n'[:PEIF_q7$$\"3C6*)4^UB'4#F6$!3Fixh+_<mIF_q7$$
\"3pJx1]UPg5F6$!33fAoTGl0JF_q7$$!3\"p&zm[Gq)G\"!#E$!35+++aEfTJF_q-%'CO
LOURG6&%$RGBG$\"#5!\"\"F(F(-%+AXESLABELSG6$Q!6\"Fg\\l-%%VIEWG6$%(DEFAU
LTGF\\]l" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cur
ve 1" }}}{PARA 11 "" 1 "" {XPPMATH 20 "6$%-inital_pointG/%#P1G6$\"\"!F
'" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%*end_pointG/%#P2G6$\"\"!,$%#PiG!
\"\"" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 72 "b) Find the equation o
f the tangent line to the curve at the given point" }}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 11 "t0 := Pi/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 
"6#>%#t0G,$*&\"\"#!\"\"%#PiG\"\"\"F*" }}}{SECT 1 {PARA 5 "" 0 "" 
{TEXT -1 8 "Solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "The equati
on of the tangent line at the given point is given by " }}{PARA 258 "
" 0 "" {TEXT -1 17 "y-y0 = m(x-x0) , " }}{PARA 0 "" 0 "" {TEXT -1 6 "w
here " }}{PARA 0 "" 0 "" {TEXT -1 34 "          m = dy/dx (t0)        \+
  " }}{PARA 0 "" 0 "" {TEXT -1 20 "          y0 = y(t0)" }}{PARA 0 "" 
0 "" {TEXT -1 20 "          x0 = x(t0)" }}{PARA 0 "" 0 "" {TEXT -1 1 "
 " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 
32 "First, find the derivative dy/dx" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "dx/dt=Diff(x(t),t);dx/dt=diff(x(t),t);dxt:=diff(x(t),
t):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#dxG\"\"\"%#dtG!\"\"-%%Diff
G6$*&%\"tGF&-%$sinG6#F-F&F-" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#dx
G\"\"\"%#dtG!\"\",&-%$sinG6#%\"tGF&*&F-F&-%$cosGF,F&F&" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "dy/dt=Diff(y(t),t);dy/dt=diff(y(t),
t); dyt:=diff(y(t),t):" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#dyG\"\"
\"%#dtG!\"\"-%%DiffG6$*&%\"tGF&-%$cosG6#F-F&F-" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/*&%#dyG\"\"\"%#dtG!\"\",&-%$cosG6#%\"tGF&*&F-F&-%$sinG
F,F&F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 40 "dy/dx = Diff('y(t
)', t)/Diff('x(t)', t);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/*&%#dyG\"
\"\"%#dxG!\"\"*&-%%DiffG6$-%\"yG6#%\"tGF0F&-F+6$-%\"xGF/F0F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "dy/dx=dyt/dxt;" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#/*&%#dyG\"\"\"%#dxG!\"\"*&,&-%$cosG6#%\"tGF&*&F.
F&-%$sinGF-F&F(F&,&F0F&*&F.F&F+F&F&F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 37 "Second, use the derivative to find m:" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 24 "m = subs(t=t0, dyt/dxt);" }}{PARA 0 "> " 0 
"" {MPLTEXT 1 0 25 "m := eval(dyt/dxt, t=t0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%\"mG*&,&-%$cosG6#,$*&\"\"#!\"\"%#PiG\"\"\"F/F/*&#F/F,
F/*&F.F/-%$sinGF)F/F/F-F/,&F3F/*&#F/F,F/*&F.F/F'F/F/F/F-" }}{PARA 11 "
" 1 "" {XPPMATH 20 "6#>%\"mG,$*&\"\"#!\"\"%#PiG\"\"\"F(" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 32 "Third, evaluate y(t0) and x(t0):" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "y0=subs(t=t0, y(t));y0 := ev
al(y(t), t=t0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#y0G,$*&#\"\"\"\"
\"#F(*&%#PiGF(-%$cosG6#,$*&F)!\"\"F+F(F(F(F(F(" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#y0G\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
42 "x0=subs(t=t0, x(t)); x0:=eval(x(t), t=t0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%#x0G,$*&#\"\"\"\"\"#F(*&%#PiGF(-%$sinG6#,$*&F)!\"\"F+
F(F(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#x0G,$*&\"\"#!\"\"%#Pi
G\"\"\"F*" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 10 "Therefore " }}}}
{SECT 1 {PARA 5 "" 0 "" {TEXT -1 6 "Answer" }}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 36 "The equation of the tangent line is:" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 16 "y-y0 = m*(x-x0);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%\"yG,$*(\"\"#!\"\"%#PiG\"\"\",&%\"xGF**&F'F(F)F*F(F*F
(" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 3 "or " }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 24 "y-y0 = expand(m*(x-x0));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%\"yG,&*&#\"\"\"\"\"#F(*&%#PiGF(%\"xGF(F(!\"\"*&#F(\"
\"%F(*$)F+F)F(F(F(" }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 96 "c) Set u
p the integral representing the surface area when revolving the curve \+
around the y-axis " }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 15 "Solution/An
swer" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "SA=2*Pi*Int(x(t)*sqr
t((Diff(x(t),t))^2+(Diff(y(t),t))^2),t=0..Pi);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%#SAG,$*(\"\"#\"\"\"%#PiGF(-%$IntG6$*(%\"tGF(-%$sinG6#
F.F(,&*$)-%%DiffG6$*&F.F(F/F(F.F'F(F(*$)-F66$*&F.F(-%$cosGF1F(F.F'F(F(
#F(F'/F.;\"\"!F)F(F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "SA=
2*Pi*int(x(t)*sqrt((diff(x(t),t))^2+(diff(y(t),t))^2),t=0..Pi);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#SAG,$*(\"\"#\"\"\"%#PiGF(-%$intG6$*
(%\"tGF(-%$sinG6#F.F(,&*$),&F/F(*&F.F(-%$cosGF1F(F(F'F(F(*$),&F7F(*&F.
F(F/F(!\"\"F'F(F(#F(F'/F.;\"\"!F)F(F(" }}}}}{SECT 1 {PARA 4 "" 0 "" 
{TEXT -1 36 "d) Evaluate the integral found in c)" }}{SECT 1 {PARA 5 "
" 0 "" {TEXT -1 8 "Solution" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
20 "SA = expand(rhs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#SAG,$*(
\"\"#\"\"\"%#PiGF(-%$intG6$*(%\"tGF(-%$sinG6#F.F(,**$)F/F'F(F(*&)F.F'F
()-%$cosGF1F'F(F(*$F7F(F(*&F6F(F4F(F(#F(F'/F.;\"\"!F)F(F(" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "simplify(%);" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%#SAG,$*(\"\"#\"\"\"%#PiGF(-%$intG6$*(%\"tGF(-%$sinG6#
F.F(,&*$)F.F'F(F(F(F(#F(F'/F.;\"\"!F)F(F(" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 21 "Evaluate the integral" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT 
-1 6 "Answer" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "SA = evalf(r
hs(%));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#SAG$\"+I'\\AB%!\")" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 83 "The surface area of the solid obta
ined by revolving the curve around the y-axis is " }}{PARA 0 "" 0 "" 
{TEXT -1 23 "SA = 42.32 units sqared" }}}}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 73 "_________________________________________________________
________________" }}}}{SECT 0 {PARA 3 "" 0 "" {TEXT -1 24 "Q.2. Consid
er the curves" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "r1:=3*sin(t
heta); r2:=3*sin(2*theta);0<= theta, theta<2*Pi;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%#r1G,$*&\"\"$\"\"\"-%$sinG6#%&thetaGF(F(" }}{PARA 11 
"" 1 "" {XPPMATH 20 "6#>%#r2G,$*&\"\"$\"\"\"-%$sinG6#,$*&\"\"#F(%&thet
aGF(F(F(F(" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$1\"\"!%&thetaG2F%,$*&\"
\"#\"\"\"%#PiGF*F*" }}}{SECT 1 {PARA 4 "" 0 "" {TEXT -1 30 "a) Sketch \+
and label the curves" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 15 "Solution/
Answer" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 76 "plot(\{[r2, theta,
 theta=0..2*Pi],[r1, theta, theta=0..2*Pi]\}, coords=polar);" }}{PARA 
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>&**\\\"F9$\"3rF5o?!)*z[\"F97$$!3+(oNtP^u[\"F9$\"37uo\"fy!Q18F97$$!3rm
I&yaTIX\"F9$\"3S46USahF6F97$$!3)es=JqFtQ\"F9$\"3yiq)H\"3I'H*F-7$$!3_P
\\V3?y%H\"F9$\"3y!zgkfgnU(F-7$$!3S]o^p'G2>\"F9$\"3>3Wn')3pxeF-7$$!3D6_
)RTY#o5F9$\"3*=4]RA2)pWF-7$$!3#eZ3'eG%\\>*F-$\"37whTg?r[JF-7$$!3mFqvD=
nXvF-$\"3FF7b^[4O?F-7$$!3yflqaMPUeF-$\"3nB\"*p.&\\X=\"F-7$$!3\"pFqjx7
\\/%F-$\"3[^G?DgpcbF07$$!3![uGq%4XT?F-$\"3KTzwVpm&R\"F07$$\"3ttR=U6ChC
Ff[m$\"3[!pfm)fB>?Fi[m-F[\\m6&F]\\mF(F^\\mF(-%%VIEWG6$%(DEFAULTGFijn" 
1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Curve 1" "Cur
ve 2" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 18 "r1 is the circle, " }}
{PARA 0 "" 0 "" {TEXT -1 22 "r2 is the 4 leaf rose." }}{PARA 0 "" 0 "
" {TEXT -1 88 "They both start and end at (0,0) but the circle is trac
ed twice over the given interval." }}}}}{SECT 1 {PARA 4 "" 0 "" {TEXT 
-1 81 "b) Find the polar coordinates of all the points of intersection
 of the two curves" }}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Solution" }}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "First, find the angles" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "'r2'='r1';" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%#r2G%#r1G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
10 "'r2-r1'=0;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&%#r2G\"\"\"%#r1G!
\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "r2-r1=0;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*&\"\"$\"\"\"-%$sinG6#,$*&\"\"#F'%&
thetaGF'F'F'F'*&F&F'-F)6#F.F'!\"\"\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 10 "expand(%);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&*(\"
\"'\"\"\"-%$sinG6#%&thetaGF'-%$cosGF*F'F'*&\"\"$F'F(F'!\"\"\"\"!" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 10 "factor(%);" }}{PARA 11 "" 1 
"" {XPPMATH 20 "6#/,$*(\"\"$\"\"\"-%$sinG6#%&thetaGF',&*&\"\"#F'-%$cos
GF*F'F'F'!\"\"F'F'\"\"!" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "
sin(theta)=0, cos(theta)=1/2;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/-%$s
inG6#%&thetaG\"\"!/-%$cosGF&#\"\"\"\"\"#" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 28 "Theta=\{solve(r2=r1, theta)\};" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#/%&ThetaG<&\"\"!%#PiG,$*&\"\"$!\"\"F'\"\"\"F,,$*&F*F+F'
F,F+" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 70 "Note 1:  -Pi/3 is not in \+
the domain, so we need to shift it up by 2Pi:" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 17 "theta=-Pi/3+2*Pi;" }}{PARA 11 "" 1 "" {XPPMATH 
20 "6#/%&thetaG,$*(\"\"&\"\"\"\"\"$!\"\"%#PiGF(F(" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 114 "Note 2: For this angle the radius is negative so \+
we can represent it alternatively as an angle in the II quadrant " }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "theta= 5/3*Pi-Pi;" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#/%&thetaG,$*(\"\"#\"\"\"\"\"$!\"\"%#PiGF(F(
" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 22 "Second, find the radii" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "'r1'=eval(r1, theta=0), eval
(r2, theta=0)='r2';" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 53 "'r1'=eval(r1
, theta=Pi/3), eval(r2, theta=Pi/3)='r2';" }}{PARA 0 "> " 0 "" 
{MPLTEXT 1 0 49 "'r1'=eval(r1, theta=Pi), eval(r2, theta=Pi)='r2';" }}
{PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "'r1'=eval(r1, theta=5/3*Pi), eval(r
2, theta=5/3*Pi)='r2';" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%#r1G\"\"!/
F%%#r2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%#r1G,$*(\"\"$\"\"\"\"\"#!
\"\"F'#F(F)F(/F%%#r2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%#r1G\"\"!/F
%%#r2G" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%#r1G,$*(\"\"$\"\"\"\"\"#!
\"\"F'#F(F)F*/F%%#r2G" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 6 "Answer
" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 60 "The polar coordinates of all t
he points of intersection are:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 160 "P1(eval(r1, theta=0), 0); P2(eval(r1, theta=Pi/3), Pi/3);P3=(
eval(r1, theta=5/3*Pi), 5/3*Pi), P3 = (eval(r1, theta=2/3*Pi), 2/3*Pi)
;P4=(eval(r1, theta=Pi), Pi);" }}{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#P1G6$\"\"!F&" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#-%#P2G6$,$*(\"\"$\"\"\"\"\"#!\"\"F(#F)F*F),$*&F(F+%#PiG
F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$/%#P3G6$,$*(\"\"$\"\"\"\"\"#!
\"\"F(#F)F*F+,$*(\"\"&F)F(F+%#PiGF)F)/F$6$,$*(F(F)F*F+F(F,F),$*(F*F)F(
F+F0F)F)" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/%#P4G6$\"\"!%#PiG" }}}}}
{SECT 1 {PARA 4 "" 0 "" {TEXT -1 79 "c) Set up an integral representin
g the area of the region common to both curves" }}{SECT 1 {PARA 5 "" 
0 "" {TEXT -1 8 "Solution" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 95 "Note1
: Since the area is symmetric we can calculate just the common area in
 the first quadrant " }}{PARA 0 "" 0 "" {TEXT -1 42 "and then double i
t to find the total area." }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 115 "Not
e2: The common area is that under the circle in (0, Pi/3) and under th
e rose in (Pi/3, Pi/2), as depicted below:" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 133 "plot(\{[r2, theta, theta=0..Pi/2],[r1 ,theta, thet
a=0..Pi/2], [r, Pi/3, r=0..3*sqrt(3)/2]\}, coords=polar, colour=[green
, green, red]);" }}{PARA 13 "" 1 "" {GLPLOT2D 220 326 326 {PLOTDATA 2 
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%pkkt*F-$\"3&4,,P:0ko\"FI7$$\"3/_&Rw<w4+\"FI$\"3h$zPofTPt\"FI7$$\"3O)o
3G'fpF5FI$\"3:j!=Bi@+y\"FI7$$\"3'f'*>;5pc0\"FI$\"3ssj)*>DZG=FI7$$\"3FP
;)*e0h#3\"FI$\"3+Qn'H\\O^(=FI7$$\"3!G&[P!4i,6\"FI$\"3G;h;Xr&G#>FI7$$\"
3Ckw<\"[&[P6FI$\"3&ekrik#=q>FI7$$\"33byiRGfi6FI$\"3G?7vm)pO,#FI7$$\"3i
@dx0(o8>\"FI$\"3JE6&*G6^j?FI7$$\"3'R'z\\$=1r@\"FI$\"3EAZ-[(*33@FI7$$\"
3]5A(GqZXC\"FI$\"3&G!y\"Q&)>c:#FI7$$\"3KRBPQF\"3F\"FI$\"3NEmwHA6,AFI7$
$\"3C+++1\"Q!*H\"FI$\"3!\\/g0+++D#FI-F[[l6&F][lF^[lF(F(-%%VIEWG6$%(DEF
AULTGF]jm" 1 2 0 1 10 0 2 9 1 4 2 1.000000 45.000000 45.000000 0 0 "Cu
rve 1" "Curve 2" "Curve 3" }}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 9 "Ther
efore" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 66 "Half_A:=1/2*(Int(r
1^2, theta=0..Pi/3)+Int(r2^2,theta=Pi/3..Pi/2));" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%'Half_AG,&*&#\"\"\"\"\"#F(-%$IntG6$,$*&\"\"*F()-%$sin
G6#%&thetaGF)F(F(/F4;\"\"!,$*&\"\"$!\"\"%#PiGF(F(F(F(*&F'F(-F+6$,$*&F/
F()-F26#,$*&F)F(F4F(F(F)F(F(/F4;F8,$*&F)F;F<F(F(F(F(" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 6 "Hence " }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 6 "
Answer" }}{EXCHG {PARA 0 "" 0 "" {TEXT -1 36 "The common area integral
 is given by" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "A:= 2*Half_
A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG,&-%$IntG6$,$*&\"\"*\"\"\"
)-%$sinG6#%&thetaG\"\"#F,F,/F1;\"\"!,$*&\"\"$!\"\"%#PiGF,F,F,-F'6$,$*&
F+F,)-F/6#,$*&F2F,F1F,F,F2F,F,/F1;F6,$*&F2F9F:F,F,F," }}}}}{SECT 1 
{PARA 4 "" 0 "" {TEXT -1 36 "d) Evaluate the integral found in c)" }}
{SECT 1 {PARA 5 "" 0 "" {TEXT -1 8 "Solution" }}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 56 "A=(int(r1^2, theta=0..Pi/3)+int(r2^2,theta=Pi/3..P
i/2));" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#/,&-%$IntG6$,$*&\"\"*\"\"\")
-%$sinG6#%&thetaG\"\"#F+F+/F0;\"\"!,$*&\"\"$!\"\"%#PiGF+F+F+-F&6$,$*&F
*F+)-F.6#,$*&F1F+F0F+F+F1F+F+/F0;F5,$*&F1F8F9F+F+F+,&*(\"#FF+\"#;F8F7#
F+F1F8*(F*F+\"\"%F8F9F+F+" }}}}{SECT 1 {PARA 5 "" 0 "" {TEXT -1 6 "Ans
wer" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 13 "'A'=evalf(A);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#/%\"AG$\"+LxuXT!\"*" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 57 "The common area to both  curves is A = 4.15 uni
ts squared" }}}}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "_________________
________________________________________________________" }}}}}{MARK "
1" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 1 }{PAGENUMBERS 0 1 2 33 1 1 }