   Chapter 15, Problem 11P

Chapter
Section
Textbook Problem

If f is continuous, show that ∫ 0 x ∫ 0 y ∫ 0 z f ( t ) d t   d z   d y = 1 2 ∫ 0 x ( x − t ) 2 f ( t )   d t

To determine

To show: 0x0y0zf(t)dtdzdy=120x(xt)2f(t)dt.

Explanation

Given:

The region E is, E={(t,z,y)|0tz,0zy,0yx}.

Calculation:

In the right hand side of the given statement the integral is present over t. So, it is necessary to rewrite the equation in left hand side in the order of z,y and t. For that project the given equation into the yt-plane. That yields, z=y and z=t yields y=t. Thus, z varies from t to y, y varies from t to x and t varies from 0 to x. Hence, the equation becomes,

0x0y0zf(t)dtdzdy=0xtxtyf(t)dzdydt

Now integrate this with respect to z and y and apply the limit.

0x0y0zf(t)dtdzdy=0xtxf(t)[z]tydydt=0xtxf(t)

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