   Chapter 15, Problem 25RE

Chapter
Section
Textbook Problem

Calculate the value of the multiple integral.25. ∬ D y   d A , where D is the region in the first quadrant bounded by the parabolas x = y2 and x = 8 − y2

To determine

To calculate: The value of given double integral over the region R.

Explanation

Given:

The region D lies in the first quadrant bounded by the parabolas x=y2 and x=8y2 .

Calculation:

From the given conditions, it is observed x varies from y2 to 8y2 and y varies from 0 to 2.

First, compute the integral with respect to x.

DydA=02[y28y2ydx]dy=02[yx]y28y2dy

Apply the limit value for x,

DydA=02y[(8y2)y2]dy=02y[8y2y2]dy

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