   Chapter 15.2, Problem 58E

Chapter
Section
Textbook Problem

Express D as a union of regions of type I or type II and evaluate the integral.58. ∬ D y   d A To determine

To express: The region D as a union of regions of type 1 or type 2 and find the value of given double integral.

Explanation

Region of type 1:

A plane region D is said to be of type 1 if it lies between two continuous functions of x.

That is, D={(x,y)|axb,g1(x)yg2(x)} ,where g1(x) and g2(x) are the continuous functions of x.

Region of type 2:

A plane region D is said to be of type 2 if it lies between two continuous functions of y.

That is, D={(x,y)|ayb,h1(y)xh2(y)} ,where h1(y) and h2(y) are the continuous functions of y.

Calculation:

The given region D as a type 2 region is split into two parts. In first part, x varies from 1 to yy3 and y varies from 1 to 0. In second part, x varies from y1 to yy3 and y varies from 0 to 1. So, the value of the double integral is,

Df(x,y)dA=101yy3ydxdy+01y1yy3ydxdy

First, compute the integral with respect to x.

101yy3ydxdy+01y1yy3ydxdy=10[xy]1yy3dy+01[xy]y1yy3dy

Apply the limit value for x,

101yy3ydxdy+01y1yy3ydxdy=1

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