Calculus: Analytic Geometry and Calculus, with Vectors

708 Iterated and multiple integrals

can advantageously be expressed in terms of spherical coordinates r, 0, 8.

When we use spherical coordinates, the set S is partitioned into subsets

S1, S2, , S by spheres having spherical equations r = ro, r = r,,

r = r., by half-planes having spherical equations ¢ _ 00, 4 01,

¢ _ 0.,, and by half-cones having the spherical equations 0 = 00,

0 = 01j , 0 = Figure 13.82 shows a typical subset containing

Figure 13.82

a point having spherical coordinates (r,4,0). When r > 0 and the num-

bers Ar, Ag, A0 are all small, this subset closely approximates a rectangu-

lar parallelepiped one dimension of which is Ar, the difference of the radii

of two spheres. The inner (or outer) face perpendicular to the ray from

the origin to the point (r,4i,6) closely approximates a rectangle one side

of which has length r A0 (the length of the arc of a sector having radius

r and central angle A0) and the perpendicular sides of which have length

r sin 0 A¢ (the length of the arc of a sector having radius r sin 0 and cen-

tral angle A4). Thus the area of the face is approximately r2 sin 0 A¢ A0.

Thus we use the formula

(13.83) AS = r2 sin 0 A4) AO Ar

(which, depending upon the choice of r, 0, 0, is exactly or approximately

correct) to put (13.81) in the form

(13.84) f f fs f(r,4,0)r2 sin0 d4) d0 dr = lim I f(r,0,¢)r2 sin 0 A4) A0 Ar.

Assuming that f is bounded over S and that f is sufficiently continuous

to make all of the integrals exist, we can use TheQrem 13.63 to express