James Stewart-Calculus_ Early Transcendentals-Cengage Learning (2015)

1064 chapter 15 Multiple Integrals
59. Suppose that f is continuous on a disk that contains the
point sa, bd. Let D r be the closed disk with center sa, bd and
radius r. Use the Mean Value Theorem for double integrals (see
Exercise 58) to show that
lim
r l 0
60. (a) Evaluate y
D
1
y
r 2 Dr
y f sx, yd dA − f sa, bd
1
sx 2 1 y 2 ny2
dA, where n is an integer and D
d
is the region bounded by the circles with center the origin
and radii r and R, 0 , r , R.
(b) For what values of n does the integral in part (a) have a
limit as r l 0 1 ?
1
y
dV, where E is the region
sx 2 1 y 2 1 z 2 ny2
d
(c) Find y y
E
bounded by the spheres with center the origin and radii r
and R, 0 , r , R.
(d) For what values of n does the integral in part (c) have a
limit as r l 0 1 ?
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