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Homework 13
APPM 2350, 2010
1. Using Green’s theorem (1), we can choose
α
(
x, y
)=0and
β
(
x, y
)=
x
so that
Area
=
±±
R
dA
=
²
C
xdy
Using the given parametrization of the deltoid, we have
²
C
xdy
=
±
t
=2
π
t
=0
(2 cos
t
+cos(2
t
))(2 cos
t

2cos(2
t
))
dt
=
±
2
π
0
(4 cos
2
t

4cos
t
cos(2
t
)+4cos
t
cos(2
t
)

2cos
2
(2
t
))
dt
=
±
2
π
0
(2(1 + cos(2
t
))

(1 + cos(4
t
))
dt
=
±
2
π
0
(1 + 2 cos(2
t
)

cos(4
t
))
dt
=2
π
2. We assume that the density of the object
ρ
is constant. The
x
and
y
centers of mass are given by
x
=
³
R
xρdA
³
R
ρdA
y
=
³
R
yρdA
³
R
ρdA
=
³
R
xdA
³
R
dA
=
³
R
ydA
³
R
dA
=
³
R
xdA
A
=
³
R
ydA
A
Recall Green’s theorem: For continuously di±erentiable
α
(
x, y
)and
β
(
x, y
)wehave
±±
R
´
∂β
∂x

∂α
∂y
µ
dA
=
²
C
αdx
+
βdy
(1)
Consider computing
x
.W
ec
h
o
o
s
e
α
(
x, y
)and
β
(
x, y
)suchtha
tthed
i±e
renceo
fpa
r
t
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 Fall '07
 ADAMNORRIS

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