Math 23
B. Dodson
Week 5a Homework:
15.5 Applications of Double Integrals
15.6
Triple integrals
15.7
Cylindrical Coordinates;
15.8
Spherical Coordinates
Problem 15.5.6:
Find the mass and center of mass of a thin
plate (lamina) occupying the triangular region
with verticies at (0
,
0)
,
(1
,
1) and (4
,
0)
,
if the density at (
x, y
) is
ρ
(
x, y
) =
x.
Solution:
We have mass
m
=
integraldisplayintegraldisplay
D
ρ
(
x, y
)
dA,
and need an iterated integral to evaluate the
double integral. Checking the sketch, we see
that the region is of Type II, with 0
≤
y
≤
1
,
and
y
≤
x
≤
4

3
y
(where the line from (1
,
1)
to (4
,
0) has slope

1
3
,
and we solve
y

0 =

1
3
(
x

4) for
x
). The iterated integral
is then
integraldisplay
1
0
integraldisplay
4

3
y
y
x dxdy.
Evaluation gives
m
=
10
3
.
For the center of mass, (
x,
y
)
,
we have
x
=
1
m
integraldisplayintegraldisplay
D
xρ
(
x, y
)
dA,
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2
and
y
=
1
m
integraldisplayintegraldisplay
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 Spring '06
 YUKICH
 Calculus, Integrals, Multiple integral, 1 M, Iterated Integrals, B. Dodson

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