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{SECT 0 {EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "Green's Theorem and the Pl
animeter" }}{PARA 0 "" 0 "" {TEXT -1 28 "Dale Winter and Oliver Knill
" }}{PARA 0 "" 0 "" {TEXT -1 7 "5/24/01" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "
" 0 "" {TEXT -1 46 "Part 1:  Iterated Integrals and Line Integrals" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 21 "restart: with(plots):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
41 "plot3d(sin(x*y),x=-1*Pi..Pi,y=-1*Pi..Pi);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 43 "int(int(sin(x*y),y=-1*Pi..Pi),x=-1*Pi..Pi);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 52 "Enter your prediction for \+
the value of the integral " }{XPPEDIT 18 0 "Int(Int(sqrt(x^2+y^2),y = \+
-sqrt(1-x^2) .. sqrt(1-x^2)),x = -1 .. 1);" "6#-%$IntG6$-F$6$-%%sqrtG6
#,&*$%\"xG\"\"#\"\"\"*$%\"yGF.F//F1;,$-F)6#,&F/F/*$F-F.!\"\"F9-F)6#,&F
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" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 38 "plot3d(sqrt(x^2+y^2),x=
-1..1,y=-1..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "int(int(
sqrt(x^2+y^2),y=-1*sqrt(1-x^2)..sqrt(1-x^2)),x=-1..1);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalf(%);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 
0 "" 0 "" {TEXT -1 59 "Set up and evaluate the integral of ln(1+xy) on
 the region " }{TEXT 257 2 "G." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
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{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 44 "Now we turn our attention
 to line integrals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 85 "gamma_1:=t->piecewise(0<=t and t<1,t,1<=t and t<2,1,2<=t and t<3
,3-t,3<=t and t<4,0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "ga
mma_2:=t->piecewise(0<=t and t<1,0,1<=t and t<2,t-1,2<=t and t<3,1,3<=
t and t<4,4-t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "P:=(x,y)
->x^2;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Q:=(x,y)->y^2;" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 100 "int(P(gamma_1(t),gamma_2(
t))*diff(gamma_1(t),t)+Q(gamma_1(t),gamma_2(t))*diff(gamma_2(t),t),t=0
..4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 73 "Express the lin
e integral as a sum of one-dimensional definite integrals." }}{PARA 0 
"" 0 "" {TEXT -1 0 "" }}}{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 -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 66 "Set up and evaluate the iterated integrals and th
e line integrals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 52 "Case 1:  The curve/boundary of region is the squar
e." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 85 "gamma_1
:=t->piecewise(0<=t and t<1,t,1<=t and t<2,1,2<=t and t<3,3-t,3<=t and
 t<4,0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "gamma_2:=t->pie
cewise(0<=t and t<1,0,1<=t and t<2,t-1,2<=t and t<3,1,3<=t and t<4,4-t
);" }}}{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 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 
0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 28 "Summarize your resul
ts here:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 73 "Case 2:  The curve/boundary of reg
ion is the circle (x-1)^2 + (y-1)^2 = 1" }}{PARA 0 "" 0 "" {TEXT -1 0 
"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 21 "gamma_1:=t->1+cos(t);" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 21 "gamma_2:=t->1+sin(t);" }}}{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 "
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"" 0 "" {TEXT -1 28 "Summarize your results here:" }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 110 "State your conjecture on the r
elationship between the values of the iterated integrals and the line \+
integrals." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 62 "Pa
rt 2:  A Link Between Line Integrals and Iterated Integrals." }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "
" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 69 "The functions and curve
s to check with the theorem are defined below." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "gamma_1:=t->cos(t);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "gamma_2:=t->sin(t);" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "P:=(x,y)->sqrt(x^2+y^2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "Q:=(x,y)->sqrt(x^2+y^2);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "The comman
ds given below produce a plot of the polar curve:  r^2 = 9*cos(theta) \+
for 0 <= theta <= 2*Pi." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 59 "alpha:=plot(3*sqrt(cos(
theta)),theta=0..2*Pi,coords=polar):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 59 "beta:=plot(-3*sqrt(cos(theta)),theta=0..2*Pi,coords=p
olar):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "display([alpha,be
ta]);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 16 "gamma_1:=t->???;" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 16 "gamma_2:=t->???;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "P:=(x,y)->cos(y);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 17 "Q:=(x,y)->sin(y);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 71 "gamma_1:=t->piecewise(0<=t a
nd t<1,1+t,t<=1 and t<2,t,t<=2 and t<=3,2);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 87 "gamma_2:=t->piecewise(0<=t and t<1,1+t,t<=1 and t<
2,1+2*(t-1),t<=2 and t<=3,2+2*(t-2));" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "P:=(x,y)->sqrt(x^2+y^2);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "Q:=(x,y)->sqrt(x^2+y^2);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 103 "The command given below produces \+
a plot of the polar curve:  r = 1 + cos(theta) for 0 <= theta <= 2*Pi.
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 46 "plot(1+cos(theta),theta=0..2*Pi,coords=polar);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 16 "gamma_1:=t->???;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "gamma_2:=t->???;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "P:=(x,y)->sqrt(x^2+y^2);" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 24 "Q:=(x,y)->sqrt(x^2+y^2);" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "Se
t up and evaluate the integral using a line integral approach and an i
terated integral approach. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 31 "Case 1:  Line integral approach" }
}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 16 "gamma_1:=t->???;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "g
amma_2:=t->???;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "P:=(x,y)
->cos(x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "Q:=(x,y)->sin(
y);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 30 "The \+
line integral is equal to:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 35 "Case 2:  Iterated integral approach" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{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 -1 34 "The iterated integral is
 equal to:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 99 "Set up an
d evaluate the integral using a line integral approach and an iterated
 integral approach. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 31 "Case 1:  Line integral approach" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{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 -1 30 "The line in
tegral is equal to:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 35 "C
ase 2:  Iterated integral approach" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "alpha:=plot(5,theta=0..2*Pi
,coords=polar):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 41 "beta:=pl
ot(1,theta=0..2*Pi,coords=polar):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 22 "display([alpha,beta]);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 55 "int(int
(x-y,y=-1*sqrt(25-x^2)..sqrt(25-x^2)),x=-5..-1);" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 50 "int(int(x-y,y=sqrt(1-x^2)..sqrt(25-x^2)),x=-1
..1);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "int(int(x-y,y=???.
.???),x=???..???);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 36 "int(i
nt(x-y,y=???..???),x=???..???);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 34 "The iterated integral \+
is equal to:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{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 -1 45 "What are some potential uses for the Theorem?" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{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 -1 0 "" }}
{PARA 0 "" 0 "" {TEXT -1 76 "Examining the validity of the Theorem for
 a rectangular region of the plane." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "assume(a>0):assume(b>a):as
sume(c>0):assume(d>c):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 104 "
gamma_1:=t->piecewise(0<=t and t<=1,a+t*(b-a),1<t and t<=2,b,2<t and t
<=3,b+(2-t)*(b-a),3<t and t<=4,a);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 108 "gamma_2:=t->piecewise(0<=t and t<=1,c,1<t and t<=2,c
+(t-1)*(d-c),2<t and t<=3,d,3<t and t<=4,d+(3-t)*(d-c));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 14 "P:=(x,y)->???;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 
1 0 14 "Q:=(x,y)->???;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "M:=(x,y)->diff(P(x,y),y);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 28 "W:=(x,y)->-1*diff(Q(x,y
),x);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 54 "int(P(gamma_1(t),gamma_2(t))*diff(gamma_1
(t)),t=0..4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "int(int(M(
x,y),y=c..d),x=a..b);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "int(Q(gamma_1(t),gamma_2(t
))*diff(gamma_2(t)),t=0..4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 31 "int(int(W(x,y),y=c..d),x=a..b);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 64 "Based on your results, what do you
 conclude about the integrals?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{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 -1 87 "Do you expect these relationships \+
to hold for curves other than rectangles and squares?" }}{PARA 0 "" 0 
"" {TEXT -1 0 "" }}}{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 -1 0 "" }}{PARA 0 
"" 0 "" {TEXT -1 39 "Part 3:  Calculating Areas in the Plane" }}{PARA 
0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "gamma_1:=t->r*cos(t);" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "gamma_2:=t->r*sin(t);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 45 "int(gamma_1(t)*diff(gamma_2(t),t),t=0..2*Pi);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "int(-1*gamma_2(t)*diff(gamma
_1(t),t),t=0..2*Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 77 "int
(gamma_1(t)*diff(gamma_2(t),t)-1*gamma_2(t)*diff(gamma_1(t),t),t=0..2*
Pi);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
85 "gamma_1:=t->piecewise(0<=t and t<1,t,1<=t and t<2,1,2<=t and t<3,3
-t,3<=t and t<4,0);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 87 "gamm
a_2:=t->piecewise(0<=t and t<1,0,1<=t and t<2,t-1,2<=t and t<3,1,3<=t \+
and t<4,4-t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 42 "int(gamma_1(t)*diff(gamma_2(
t),t),t=0..4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "int(-1*ga
mma_2(t)*diff(gamma_1(t),t),t=0..4);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 74 "int(gamma_1(t)*diff(gamma_2(t),t)-1*gamma_2(t)*diff(g
amma_1(t),t),t=0..4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }
}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 
"" {MPLTEXT 1 0 77 "gamma_1:=t->piecewise(0<=t and t<1,-1+2*t,1<=t and
 t<2,2-t,2<=t and t<3,2-t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
72 "gamma_2:=t->piecewise(0<=t and t<1,0,1<=t and t<2,t-1,2<=t and t<3
,3-t);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 42 "int(gamma_1(t)*diff(gamma_2(t),t),t=0..3)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "int(-1*gamma_2(t)*diff
(gamma_1(t),t),t=0..3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "
int(gamma_1(t)*diff(gamma_2(t),t)-1*gamma_2(t)*diff(gamma_1(t),t),t=0.
.3);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
104 "gamma_1:=t->piecewise(0<=t and t<=1,a+t*(b-a),1<t and t<=2,b,2<t \+
and t<=3,b+(2-t)*(b-a),3<t and t<=4,a);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 108 "gamma_2:=t->piecewise(0<=t and t<=1,c,1<t and t<=2,c
+(t-1)*(d-c),2<t and t<=3,d,3<t and t<=4,d+(3-t)*(d-c));" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 42 "int(gamma_1(t)*diff(gamma_2(t),t),t=0..4);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "int(-1*gamma_2(t)*diff(gamma
_1(t),t),t=0..4);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 74 "int(ga
mma_1(t)*diff(gamma_2(t),t)-1*gamma_2(t)*diff(gamma_1(t),t),t=0..4);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 185 "List yo
ur results from evaluating the integrals here.  What similarities do y
ou see in the values of the integrals that you have calculated?  How c
an you account for these similarities?" }}{PARA 0 "" 0 "" {TEXT -1 0 "
" }}}{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 -1 113 "What do these line integrals
 appear to be actually calculating?  Summarize your findings by statin
g a conjecture." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{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 -1 98 "Test your conjecture by defining several curves and \+
evaluating the line integrals on those curves." }}{PARA 0 "" 0 "" 
{TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 163 "NOTE:  The limits of int
egration will depend on how you define the curves.  Don't forget to en
ter appropriate limits of integration as well as defining the curves.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "gamma_1
:=t->???;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "gamma_2:=t->??
?;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 46 "int(gamma_1(t)*diff(gamma_2(t),t),t=???..???)
;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "int(-1*gamma_2(t)*diff
(gamma_1(t),t),t=???..???);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
78 "int(gamma_1(t)*diff(gamma_2(t),t)-1*gamma_2(t)*diff(gamma_1(t),t),
t=???..???);" }}}{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 -1 111 "Use your con
jecture to find the area enclosed by the curve (x^2)/(a^2) + (y^2)/(b^
2) = 1 using a line integral." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 16 "gamma_1:=t->???;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "gamma_2:=t->???;" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "int(???
,t=???..???);" }}}{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 -1 76 "Supply a ma
thematical argument to verify the correctness of your conjecture." }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{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 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" 
{TEXT -1 41 "Part 4:  Experiments with the Planimeter." }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 68 "What is the ratio for the area of \+
a cicle over the area of a square?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" 
{TEXT -1 217 "List the measurements that you made using the planimeter
 for the circle and the square.  What is the ration of these measureme
nts?  What property of a region of the plane do you think the planimet
er actually measures?" }}{PARA 0 "" 0 "" {TEXT -1 2 "  " }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT 
-1 134 "What is your equation for converting the output of the planime
ter?  What does your equation convert the output of the planimeter int
o?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 2 "  " }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 116 "List the p
lanimeter results for the triangle and ellipse.  Do these validate the
 conversion equation you have found?" }}{PARA 0 "" 0 "" {TEXT -1 0 "" 
}}}{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 -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 42 "
Part 5:  The Mathematics of the Planimeter" }}{PARA 0 "" 0 "" {TEXT 
-1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 
0 "" 0 "" {TEXT -1 42 "Graphical Representations of Vector Fields" }}
{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 
0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 56 "a:=(x,y)->(x^3+x*y
^2+sqrt(4*x^2*y^2+y^6))/(2*(x^2+y^2)):" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 56 "b:=(x,y)->(x^2*y+y^3-sqrt(x^6+4*x^2*y^2))/(2*(x^2+y^2
)):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "P:=(x,y)->y^2+b(x,y)
:" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "Q:=(x,y)->2*x-a(x,y):
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "fieldplot([P(x,y),Q(x,y
)],x=-2..2,y=-2..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}
}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "a:=(x,y)->(x^3+x*y^2+sqrt(4
*x^2*y^2-x^4*y^2+4*y^4-2*x^2*y^4-y^6))/(2*(x^2+y^2)):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 80 "b:=(x,y)->(x^2*y+y^3-sqrt(4*x^4-x^6
+4*x^2*y^2-2*x^4*y^2-x^2*y^4))/(2*(x^2+y^2)):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 22 "P:=(x,y)->-1*y+b(x,y):" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 19 "Q:=(x,y)->x-a(x,y):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 43 "fieldplot([P(x,y),Q(x,y)],x=-2..2,y=-2..2);" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 64 "a:=(x,y)->sqrt(4*x^2*y^2-x^4*y^2+4*y^4-2*x^2*y^4-y^
6)/(x^2+y^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 67 "b:=(x,y)->
-1*sqrt(4*x^4-x^6+4*x^2*y^2-2*x^4*y^2-x^2*y^4)/(x^2+y^2):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "P:=(x,y)->y+b(x,y):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "Q:=(x,y)->x+a(x,y):" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 43 "fieldplot([P(x,y),Q(x,y)],x=-2..2,y
=-2..2);" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "a:=(x,y)->(x^3+x*y^2+sqrt(4*x^2*y^2
+4*y^4))/(x^2+y^2):" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "b:=(
x,y)->(x^2*y+y^3-sqrt(4*x^4+4*x^2*y^2))/(x^2+y^2):" }}}{EXCHG {PARA 0 
"> " 0 "" {MPLTEXT 1 0 22 "P:=(x,y)->-1*y+b(x,y):" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 22 "Q:=(x,y)->-1*x-a(x,y):" }}}{EXCHG {PARA 0 ">
 " 0 "" {MPLTEXT 1 0 43 "fieldplot([P(x,y),Q(x,y)],x=-2..2,y=-2..2);" 
}}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 
0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 78 "What is \+
your interpretation of the line integral of F(x,y) around the curve C?
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 118 "Solve for the coordinates of the \+
elbow (a and b) of the planimeter in terms of the coordinates of the w
heel (x and y)." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 65 "assume(a>0):assume(b>0):solvefor(\{a^2+b^2=1, \+
(x-a)^2+(y-b)^2=1\});" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "P:
=(x,y)->???;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 14 "Q:=(x,y)->?
??;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "
> " 0 "" {MPLTEXT 1 0 43 "fieldplot([P(x,y),Q(x,y)],x=-2..2,y=-2..2);
" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> \+
" 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "" 0 "" {TEXT -1 75 "Consid
er the iterated intgral.  Simplify the integrand as much as possible.
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 24 "assume(x>0):assume(y>0):" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 14 "simplify(???);" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 28 "What property of the region " }
{TEXT 256 1 "D" }{TEXT -1 39 " would the iterated integral calculate?
" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "" 0 "" {TEXT -1 167 "Look back at the conversion formu
la that you found in the previous section of the module.  How can you \+
account for the algebraic structure of your conversion equation?" }}
{PARA 0 "" 0 "" {TEXT -1 2 "  " }}}{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 
-1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "Part 6:  Summary" }}{PARA 0 "" 
0 "" {TEXT -1 0 "" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 0 "" }}}{EXCHG {PARA 0 "> " 0 "
" {MPLTEXT 1 0 0 "" }}}}{MARK "299" 0 }{VIEWOPTS 1 1 0 1 1 1803 1 1 1 
1 }{PAGENUMBERS 0 1 2 33 1 1 }