Calculus: Analytic Geometry and Calculus, with Vectors

710 Iterated and multiple integrals

order of integration in such a way that the last integration is with respect to r
and simplify the result as much as possible for the special case in which the
density is a function of r alone, say 3(r,-O,0) = f(r). Ans.:

M =

f0a

dr f7 d6

fo2r

S0)r2 sin 0 dq,, M = Oar roa r2f(r) dr.

2 Show how the last result of Problem 1 can be obtained by direct use of

spherical shells and without use of iterated integrals.

3 Solve the modification of Problem 1 in which S is a spherical shell bounded

by concentric spheres having radii ri and f2-

4 Let q be a nonnegative constant and let S be a spherical ball of radius a

whose density is proportional the qth power of the distance from the center.

Using the formula

r2 sin 0 A¢ AO Ar

for volume in spherical coordinates and taking the origin at the center of the ball,

set up and evaluate a triple integral for the polar moment of inertia of S about

the z axis. Rns.: The triple integral is obtained by setting S(r,4,0) = kra in

(13.88). The required moment is

81rkaQ+1

3(q+5)

5 Show how the preceding problem gives the conclusion that the moment

of inertia, about a diameter, of a spherical ball having radius a and uniform

density S is -rrasS.

6 A solid spherical ball of radius a has, at each point P, density equal to

the product of the distances from P to the origin and to the axis from which 0

is measured. Set up and evaluate a threefold iterated integral in spherical

coordinates for the mass M of the ball. Ans.:

M _ f02r

d4ifar sins0 d0

fa

r4 dr = i2as.

7 This problem involves lengths of curves. Suppose that, as time t increases

from a to b, a particle P moves along a curve C in such a way that its rectangular

coordinates x, y, z, its cylindrical coordinates p, q,, z, and its spherical coordinates

r, d, 0 are all functions oft having continuous derivatives. Start with the formula

(1) L=Jab \d )2+

(4y)2

+

(z)' dt

giving the length L of the curve C as an integral involving rectangular coordinates.

Use the formulas

(2) x=pcos y=psinq, z = z

to obtain the formula

(

(3) L=fab /p2\d) +( )Z+(at)dtZ

giving Las an integral involving cylindrical coordinates. Then use the formulas

(4) p=rsin0, 0=0, z=rcos0