James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

Section 15.4 Applications of Double Integrals 1019
D
y
a
3a
”0, ’
2π
≈+¥=a@
SOLUTION Let’s place the lamina as the upper half of the circle x 2 1 y 2 − a 2 . (See
Figure 6.) Then the distance from a point sx, yd to the center of the circle (the origin) is
sx 2 1 y 2 . Therefore the density function is
sx, yd − Ksx 2 1 y 2
_a
FIGURE 6
0
a
x
where K is some constant. Both the density function and the shape of the lamina
suggest that we convert to polar coordinates. Then sx 2 1 y 2 − r and the region D is
given by 0 < r < a, 0 < < . Thus the mass of the lamina is
m − y
D
y sx, yd dA − y
D
y Ksx 2 1 y 2 dA
− y
0 ya sKrd r dr d − K y
d y a
r 2 dr
0
0 0
− K r a− 3 Ka3
3
3G0
Both the lamina and the density function are symmetric with respect to the y-axis, so
the center of mass must lie on the y-axis, that is, x − 0. The y-coordinate is given by
Compare the location of the center of
mass in Example 3 with Example 8.3.4,
where we found that the center of mass
of a lamina with the same shape but
uniform density is located at the point
s0, 4ays3dd.
y − 1 y ysx, yd dA − 3
y
m Ka 3 0 ya r sin sKrd r dr d
0
D
− 3
y
a 3 0
sin d y a
− 3
a 3 2a 4
4 − 3a
2
0
r 3 dr − 3
a 3 f2cos g 0
F r 4
a
4G0
Therefore the center of mass is located at the point s0, 3ays2dd.
■
Moment of Inertia
The moment of inertia (also called the second moment) of a particle of mass m about
an axis is defined to be mr 2 , where r is the distance from the particle to the axis. We
extend this concept to a lamina with density function sx, yd and occupying a region D
by proceeding as we did for ordinary moments. We divide D into small rect angles,
approximate the moment of inertia of each subrectangle about the x-axis, and take the
limit of the sum as the number of subrectangles becomes large. The result is the moment
of inertia of the lamina about the x-axis:
6 I x − lim o n
m, nl ` om
i−1 j−1
sy ij *d 2 sx ij *, y ij *d DA − yy y 2 sx, yd dA
D
Similarly, the moment of inertia about the y-axis is
7 I y − lim o n
m, nl ` om
i−1 j−1
sx ij *d 2 sx ij *, y ij *d DA − yy x 2 sx, yd dA
D
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