   Chapter 15.2, Problem 52E

Chapter
Section
Textbook Problem

Evaluate the integral by reversing the order of integration. ∫ 0 1 ∫ x 2 1 y   sin y   d y   d x

To determine

To reverse: The order of the integration and find the value of given double integral.

Explanation

Given

The function is f(x,y)=ysiny.

The domain D is, D={(x,y)|0x1,x2y1}.

Calculation:

The upper limit of both x and y is 1. Re write the equation as,

x2=yx=y

Reverse the order of integration, D will become D={(x,y)|0xy,0y1}. The value of the double integral is,

Df(x,y)dA=01x21ysinydydx=010yysinydxdy

First, compute the integral with respect to x.

010yysinydxdy=01[0yysinydx]dy=01ysiny[x]0ydy

Apply the limit value for x,

01

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