Calculus: Analytic Geometry and Calculus, with Vectors

13.8 Triple integrals; spherical coordinates 709

this triple integral as an iterated integral. When limits of integration
for iterated integrals are to be determined, information obtained by
looking at Figure 13.82 can be helpful. Adding subsets for which r
varies (0 and 0 being fixed) yields a spike or tapered column. Adding
spikes for which 0 varies (0 being fixed) yields a whole or a part of a wedge
which in some cases looks like a conventional wedge of an orange or
lemon or cake. Adding the wedges obtained for the appropriate values
of ¢ then gives the entire solid S. Articulate persons can describe results
of performing summations and integrations in other orders. For exam-
ple, adding subsets for which 0 varies (r and 0 being fixed) yields all or
a part of a circular hoop or ring, and there are two ways in which these
hoops can be added to yield more extensive parts of S.
Supposing that S is a spherical ball having center at the origin and
radius R and that the density (mass per unit volume) at the point having
spherical coordinates (r,c,0) is 6(r,0,6), we set up an integral for the polar
moment of inertia M 0,,,.0 of the ball about the z axis. For the volume
AS of a subset of the ball, we use the formula

(13.85) AS = r2 sin 0 A¢ A0 Ar

which, like the telephone number of a dentist, is sometimes needed but

is usually not permanently remembered. To get the mass AM of the

subset, we multiply by the density (mass per unit volume) 5(r,4,@) to

obtain

(13.86) AM = sin 0 A¢ AB Ar.

Then we must be wise and strong enough to multiply this by (r sin 0)2,

the square of the distance from the subset to the z axis, to obtain the

expression

(13.861) 8(r,cb,8)r' sing 0 A4 AB Ar

for the polar moment of the subset about the z axis. Adding these and

taking the limit of the sum gives

(13.87) My2'o,,,ao = f f fs3(r,O,0)r' sins 0 do de dr.

Since S is an entire sphere with center at the origin and radius R,

(13.88) M-o,,,_o =

(o2"

d-O fox d8

fox

3(r,4,8)r' sins 0 dr.

Problems 13.89

1 Let S be a spherical ball of radius a. Supposing that the center of the

sphere is at the origin and that the density is &(r,4,O) at the point havingspherical

coordinates (r,¢,8), set up an iterated integral for the massM. Arrange the