Chapter 14.2, Problem 75E

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

Chapter
Section

### Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
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 âˆ«0aâˆ«0bemax{b2x2,a2y2}dydx, limits aÂ andÂ b are positive.

Formula used:

Calculation:

Assume I=âˆ«0aâˆ«0bemax{b2x2,a2y2}dydx.

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

In the lower triangle, b2x2â‰¥a2y2 because yâ‰¤bax.

For the lower triangle,

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

Similarly,

For the upper triangle,

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

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

The value of the integral is:

I=âˆ«0aâˆ«0bxaeb2x2dydx+âˆ«0bâˆ«0aybea2y2dxdy

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