Calculus: Analytic Geometry and Calculus, with Vectors

13.6 Triple integrals; rectangular coordinates^701

Remark: Section 13.8 will enable us to avoid this and some other unpleasant
integrals.
7 The tetrahedron (or pyramid) bounded by the planes x = 0, y = 0,
z = 0, and

has density (mass per unit volume) S(x,y,z) at the point (x,y,z). Supposing

that a, b, c are positive constants and that p is a nonnegative constant, set up two

different iterated integrals for the pth moment of the solid about the plane

x = 0. f1ns.: One of the possibilities is

1 a ( l

Mix,^1 P rb\1 a/ r1 a b/

0

z J d J S(x,y,z) dz.

0 0

8 For the case in which S(x,y,z) = 1, so that the solid is homogeneous, show

that the last formula of Problem 7 can be put in the form

111x 0 ' ffoa xp (1- a)z dx.

At least when p = 0 and p = 1, show that

MX-0 = (p^1

+ 3)1an+lbc =(p + 1) (p + 2) (p + 3)a+lbt

Finally, show that x = a/4.

9 Let 0 < b a. A spherical ball of radius a has its center at the origin.

Set up a threefold iterated integral for the volume V of the part of the ball drilled

away when a bit of radius b drills a cylindrical hole centered onthe line having the

equations x = 0, y = a - b. Symmetry may be used, and the integral need not

be evaluated. f1ns.:

Y= 4

a

d y

b=-b)1'dx rVJa'dz.

a-2b YJo J0

10 Show that when b = a, the answer to Problem 9 reduces, as itshould, to

an integral for the volume of the wholespherical ball.

11 Assuming that the spherical ball of Problem 9 has densityS(x,y,z) at

P(x,y,z), modify the integral of the preceding problem to obtain anintegral for

the amount by which the drilling of the hole decreases thepolar moment of inertia

about the z axis. f4ns.: Multiply the invisible 1 preceding dz by the factor

S(x,y,z)(x2 + y2).

12 Modify the answer to Problem 9 to obtain an integral forthe volume of the

material drilled from the ball when the hole is centered onthe line which meets

the y axis where the surface of the ball does.

13 Supposing that a > 0, evaluate

fffsl+x+Y+zdS,