   Chapter 15.2, Problem 55E

Chapter
Section
Textbook Problem

Evaluate the integral by reversing the order of integration. ∫ 0 1 ∫ arcsin y π / 2 cos   x   1 + cos 2 x   d x   d y

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)=cosx1+cos2x .

The domain D is, D={(x,y)|arcsinyxπ2,0y1} .

Calculation:

Reverse the order of integration, D will become D={(x,y)|0xπ2,0ysinx} . The value of the double integral is,

Df(x,y)dA=01arcsinyπ2cosx1+cos2xdxdy=0π20sinxcosx1+cos2xdydx

First, compute the integral with respect to y.

0π20sinxcosx1+cos2xdydx=0π2[0sinxcosx1+cos2xdy]dx=0π2cosx1+cos2x[y]0sinxdx

Apply the limit value for y,

0π20sinxcosx1+cos2xdydx=0π2cosx1+cos2x[sinx0]dx=0π2sinxcosx1+cos2xdx

Compute the integral with respect to x.

Let u=cosx .

Then, du=sinxdx .

Therefore, u varies from 1 to 0.

0π20sinxcosx1+cos2xdydx=0π2sinxcosx1+cos2xdx=10u1+u2du

Again to integrate this, substitute t=u2,dt=2udu

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