Calculus: Analytic Geometry and Calculus, with Vectors

13.8 Triple integrals; spherical coordinates 713

partitions of the ball and principles of the integral calculusto obtain

(6) F = lim I i F

and

F = Gm NS

(DI

sin 0
(7) .l J's (D2 + 2Dr cos B + r2)9h.4P dr do dB,

where the integral is a triple integral, S is the ball or the portion of £3 occupied
by the ball, and F is the total force on the particle of massm. The formula

(8) F = Gm r a r2 dr r' sin 0 2.

() o o (D2 + 2Dr cos 0 + r2)%dB fo 5(r,0,8)7P do

shows one of the six ways in which F can be represented asan iterate integral.
The first phase of our work is done, and we proceedto see how (8) can be
simplified when the density 3(r,4),8) is independent of 0 so that S(r,4),0) _
Sl(r,8), where Sl is a function of r and 0 only. In this case the last integral in
(8) is
(9)
o

2r

61(r,8)[r sin 0 cos Oi + r sin 0 sin 4)j + (D +r cos 0)k] do.

With the aid of the fact that foe'sin 4) d¢ = 0 and foe cos 0 do = 0, we see

that this reduces to

(10) 21r5j(r,6)(D + r cos 8)k

and hence that (8) reduces to

a 2i Si(r,8)(D + cos 0) sin 0

(11) F = 2irGmk f

o

r- dr f

o (D2 + 2Dr

r

cos 0 + r2)3h d8.

This shows that, when the density is independent of .0, the components of F
in the directions of the x and y axes are 0. Of course, wise scientists always claim
that this must be true "on account of symmetry."
Our final step is to make additional simplification of (11) for the case in which
the density is a function of r only, so that Sl(r,8) = 52(r) and the ball is said to
be radially homogeneous. One reason for interest in this case lies in the fact
that suns and planets and moons are closely approximated by radially homo-
geneous balls ui less rapid rotations about their axes produce nontrivial equatorial
bulges. When Sl(r,8) = 82(r), we can put (11) in the form

(12)

where

(13)

F = 21rGmk f oa r&2(r)f (r) dr

(D+rcos 8)rsin8

f(r)-fo (D2 + 2Dr cos 0 + r2)%dB.

The integral in (13) may seem to be quite impenetrable until its fundamental

weakness is discovered. If we set u = r cos 0, then (for each fixed r) du = -r