   Chapter 14.2, Problem 75E

Chapter
Section
Textbook Problem

Evaluate ∫ 0 a ∫ 0 b e max { b 2 x 2 , a 2 y 2 } d y   d x where a and b are positive.

To determine

To calculate: The value of the integral 0a0bemax{b2x2,a2y2}dydx.

Explanation

Given:

In the integral 0a0bemax{b2x2,a2y2}dydx, limits a and b are positive.

Formula used:

Calculation:

Assume I=0a0bemax{b2x2,a2y2}dydx.

Divide the rectangle into two parts by the diagonal ay=bx.

In the lower triangle, b2x2a2y2 because ybax.

For the lower triangle,

Limits of x lie from 0 to 1 and limits of y lie from 0 to bax.

Similarly,

For the upper triangle,

The limits of y lie from 0 to 1 and limits of x lie from 0 to aby.

The value of the integral is calculated by adding the integral on both the triangles.

The value of the integral is:

I=0a0bxaeb2x2dydx+0b0aybea2y2dxdy

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