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

Section 15.8 Triple Integrals in Spherical Coordinates 1045
1. Sketch carefully the solid enclosed by the three cylinders x 2 1 y 2 − 1, x 2 1 z 2 − 1, and
y 2 1 z 2 − 1. Indicate the positions of the coordinate axes and label the faces with the equations
of the corresponding cylinders.
2. Find the volume of the solid in Problem 1.
CAS
3. Use a computer algebra system to draw the edges of the solid.
4. What happens to the solid in Problem 1 if the radius of the first cylinder is different
from 1? Illustrate with a hand-drawn sketch or a computer graph.
5. If the first cylinder is x 2 1 y 2 − a 2 , where a , 1, set up, but do not evaluate, a double integral
for the volume of the solid. What if a . 1?
Another useful coordinate system in three dimensions is the spherical coordinate system.
It simplifies the evaluation of triple integrals over regions bounded by spheres or cones.
z
˙
∏
P(∏, ¨, ˙)
Spherical Coordinates
The spherical coordinates s, , d of a point P in space are shown in Figure 1, where
− | OP | is the distance from the origin to P, is the same angle as in cylindrical coordi
nates, and is the angle between the positive z-axis and the line segment OP. Note that
O
> 0
0 < <
x
¨
FIGURE 1
The spherical coordinates of a point
y
The spherical coordinate system is especially useful in problems where there is symmetry
about a point, and the origin is placed at this point. For example, the sphere with center the
origin and radius c has the simple equation − c (see Figure 2); this is the reason for
the name “spherical” coordinates. The graph of the equation − c is a vertical halfplane
(see Figure 3), and the equation − c represents a half-cone with the z­axis as its
axis (see Figure 4).
z
z
z
z
c
x
0
y
x
0
c
y
x
0
y
x
0
c
y
0<c<π/2
π/2<c<π
FIGURE 2 − c, a sphere FIGURE 3 − c, a half-plane FIGURE 4 − c, a half-cone
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