   Chapter 15, Problem 58RE

Chapter
Section
Textbook Problem

The Mean Value Theorem for double integrals says that if f is a continuous function on a plane region D that is of type I or II, then there exists a point (x0, y0) in D such that ∬ D f ( x , y )   d A = f ( x 0 , y 0 ) A ( D ) Use the Extreme Value Theorem (14.7.8) and Properly 15.2.11 of integrals to prove this theorem. (Use the proof of the single-variable version in Section 6.5 as a guide.)

To determine

To prove: The Mean value theorem for double integrals.

Explanation

Given:

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).

Formula used:

Extreme value theorem for functions of two variables:

If f is continuous on a closed, bounded set D in , then f attains an absolute maximum value f(x1,y1) and an absolute minimum value f(x2,y2) at some points (x1,y1) and (x2,y2) in D.

If mf(x,y)M for all (x,y) in D, then mA(D)Df(x,y)dAMA(D)

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