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

Section 15.8 Triple Integrals in Spherical Coordinates 1049
Figure 10 gives another look (this
time drawn by Maple) at the solid of
Example 4.
SOLUtion Notice that the sphere passes through the origin and has center s0, 0, 1 2 d. We
write the equation of the sphere in spherical coordinates as
The equation of the cone can be written as
2 − cos or − cos
cos − s 2 sin 2 cos 2 1 2 sin 2 sin 2 − sin
This gives sin − cos , or − y4. Therefore the description of the solid E in
spherical coordinates is
FIGURE 10
E − hs, , d | 0 < < 2, 0 < < y4, 0 < < cos j
Figure 11 shows how E is swept out if we integrate first with respect to , then , and
then . The volume of E is
VsEd − y y dV − y 2
y y4
y cos
2 sin d d d
0 0 0
E
− y 2
d y y4
sin F 3
0
0
3G−0
−cos
d
tec Visual 15.8 shows an animation
of Figure 11.
− 2 3 yy4 0
sin cos 3 d − 2 y4
F2 cos4
− 3 4 8
G0
z
z
z
x
y
x
y
x
y
FIGURE 11
∏ varies from 0 to cos ˙
while ˙ and ¨ are constant.
˙ varies from 0 to π/4
while ¨ is constant.
¨ varies from 0 to 2π.
■
1–2 Plot the point whose spherical coordinates are given. Then
find the rectangular coordinates of the point.
1. (a) s6, y3, y6d (b) s3, y2, 3y4d
2. (a) s2, y2, y2d (b) s4, 2y4, y3d
3–4 Change from rectangular to spherical coordinates.
3. (a) s0, 22, 0d (b) s21, 1, 2s2 d
4. (a) s1, 0, s3 d (b) ss3 , 21, 2s3 d
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