Calculus: Analytic Geometry and Calculus, with Vectors

704 Iterated and multiple integrals

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

expression

(13.763) [z2 + p2 sin2 0j3(p,4,,%)p 042 AP Az

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

taking the limit of the sum gives

(13.764) My2 o,._o= fffs [z2 +p2 sin2 .0]3(p,o,z)p do dp dz.

This is, as it should be, the sum of the moments of inertia of S about the

two planes z = 0 and y = 0. The limits of integration being determined

with the aid of Figure 13.75, we can obtain the iterated integral formula

(13.77)

=fox

do

f0R

dp

foH

[z2 + p2sin2 o]b(p,o,z)p dz.

In order to be able to evaluate this integral in decimal form, we must

know R, H, and S(p,o,z). For the special case in which q is a nonnegative

constant and S(p,c,z) = kpQ, we can evaluate the integrals in terms of

R, H, q, and k to obtain

(13.78) M=i'o,y_o = k[27rTq +2 +aH q

+

4

The result for the case in which the cylinder is homogeneous is obtained

by setting q = 0.

Problems 13.79

1 Supposing that 0 < b < a, set up and evaluate a threefold iterated integral

in cylindrical coordinates for the volume F of the solid lying inside the sphere

and cylinder having the cylindrical equations p2 + z2 = a2 and p = b. Ans.:

F =8 fOr/2d¢f bp dpfo dz = fr[a'- (a2 -

2 Supposing that 0 < b < a, set up and evaluate a threefold iterated integral

for the mass M of the solid lying inside the sphere but outside the cylinder having

the cylindrical equations p2 + z2 = a2 and p = b, it being assumed that the

density of the body at P(p,o,z) is jzj. Ans.:

M = 8 wl

d¢

(bap dpfo a=-1

z dz =2 (a2- b2)2.

3 Set up and evaluate a threefold iterated integral in cylindrical coordinates

for the volume F of the solid bounded by the sphere and cylinder having the

cylindrical equations p2 + z2 = a2 and p = a cos 0. Bns.:

P -^4

1

/or/2 doJ /oa e-,p dp f0 _ pa

dz

=/2 r 2_

-uae f0 (1 - sine4) do = 2 -j1-