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

1036 Chapter 15 Multiple Integrals
If the density is constant, the center of mass of the solid is called the centroid of E. The
moments of inertia about the three coordinate axes are
16 I x − y y sy 2 1 z 2 d sx, y, zd dV
E
I z − y ysx 2 1 y 2 d sx, y, zd dV
E
I y − y y sx 2 1 z 2 d sx, y, zd dV
E
As in Section 15.4, the total electric charge on a solid object occupying a region E
and having charge density sx, y, zd is
Q − y y sx, y, zd dV
E
If we have three continuous random variables X, Y, and Z, their joint density function
is a function of three variables such that the probability that sX, Y, Zd lies in E is
In particular,
PssX, Y, Zd [ Ed − y y f sx, y, zd dV
E
Psa < X < b, c < Y < d, r < Z < sd − y b
The joint density function satisfies
a yd c
ys f sx, y, zd dz dy dx
r
f sx, y, zd > 0
y`
2` y`
2` y`
2`
f sx, y, zd dz dy dx − 1
ExamplE 6 Find the center of mass of a solid of constant density that is bounded by the
parabolic cylinder x − y 2 and the planes x − z, z − 0, and x − 1.
z=x
z
SOLUTION The solid E and its projection onto the xy-plane are shown in Figure 16.
The lower and upper surfaces of E are the planes z − 0 and z − x, so we describe E as
a type 1 region:
E
0
E − hsx, y, zd | 21 < y < 1, y2 < x < 1, 0 < z < x j
x
1
y
Then, if the density is sx, y, zd − , the mass is
y
x=¥
D
x=1
m − y y dV − y 1 y x
21 y1 dz dx dy
y 2 0
E
− y 1 21 y1 y 2 x dx dy − y 1 21F x 2
x−1
2G
x−y 2 dy
0
x
− 2 y1 s1 2 y 4 d dy − y 1
s1 2 y 4 d dy
21
0
FIGURE 16
− Fy 2 y 5
1
5G0
− 4 5
Copyright 2016 Cengage Learning. All Rights Reserved. May not be copied, scanned, or duplicated, in whole or in part. Due to electronic rights, some third party content may be suppressed from the eBook and/or eChapter(s).
Editorial review has deemed that any suppressed content does not materially affect the overall learning experience. Cengage Learning reserves the right to remove additional content at any time if subsequent rights restrictions require it.