Page 222 - Schaum's Outline of Theory and Problems of Advanced Calculus

CHAP. 9] MULTIPLE INTEGRALS 213

Following the same pattern as with cylindrical coordinates we discover that
2
dV ¼ r sin dr d d
and the iterated triple integral of Fðr; ; Þ has the spherical representation
ð ð ð
r 2 2 ð Þ 2 ðr; Þ
2
Fðr; ; Þ r sin dr d d
r 1 1 ð Þ 1 ðr; Þ
Of course, the order of these integrations may be adapted to the geometry.
The coordinate surfaces in spherical coordinates are spheres, cones, and planes. If r is held
constant, say, r ¼ a, then we obtain the differential element of surface area
2
dA ¼ a sin d d
The first octant surface area of a sphere of radius a is
ð =2 ð =2 ð =2 ð =2
2 2 2 2 2
0
a sin d d ¼ a ð cos Þ d ¼ a d ¼ a
0 0 0 0 2
2
Thus, the surface area of the sphere is 4 a .

Solved Problems

DOUBLE INTEGRALS
2
9.1. (a) Sketch the region r in the xy plane bounded by y ¼ x ; x ¼ 2; y ¼ 1.
ðð
2
2
(b) Give a physical interpreation to ðx þ y Þ dx dy.
r
(c) Evaluate the double integral in (b).
(a) The required region r is shown shaded in Fig. 9-6 below.
2
2
(b)Since x þ y is the square of the distance from any point ðx; yÞ to ð0; 0Þ,wecan consider the double
integral as representing the polar moment of inertia (i.e., moment of inertia with respect to the origin) of
the region r (assuming unit density).

Fig. 9-6 Fig. 9-7