   Chapter 15, Problem 59RE

Chapter
Section
Textbook Problem

Suppose that f is continuous on a disk that contains the point (a, b). Let Dr be the closed disk with center (a, b) and radius r. Use the Mean Value Theorem for double integrals (see Exercise 58) to show that lim r → 0 1 π r 2 ∬ D r f ( x , y )   d A = f ( a ,   b )

To determine

To prove: The given statement limr01πr2Drf(x,y)dA=f(a,b).

Explanation

Given:

The function f is a continuous function on a disk D that contains the point (a,b).

Dr be the closed disk with center (a,b) and radius r.

Formula used:

The Mean value theorem:

If f is a continuous function on a plane region D that is of type 1 or 2, then there exists a point (x0,y0) in D such that Df(x,y)dA=f(x0,y0)A(D).

Proof:

From the mean value theorem mentioned above, it is observed that Drf(x,y)dA=f(a,b)A(D). The given region is disk with radius r. So, the area of the region is given by πr2. Thus the above equation can be rewrite as follows.

Drf(x,y)dA=f(a,b)(πr2)1πr2Drf(x,y)dA=f(a,b)1πr2Dr[f(x,y)f(a,b)]dA=0

From the above equation, it is enough to show that limr01πr2Dr[f(x,y)f(a,b)]dA=0

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