   Chapter 15.2, Problem 54E

Chapter
Section
Textbook Problem

Evaluate the integral by reversing the order of integration. ∫ 0 2 ∫ y / 2 1 y   cos ( x 3 − 1 )   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)=ycos(x31) .

The domain D is, D={(x,y)|0xy2,0y2} .

Calculation:

Reverse the order of integration, D will become D={(x,y)|0x1,0y2x} . The value of the double integral is,

Df(x,y)dA=020y2ycos(x31)dxdy=0102xycos(x31)dydx

First, compute the integral with respect to y.

0102xycos(x31)dydx=01[02xycos(x31)dy]dx=01cos(x31)[y22]02xdx

Apply the limit value for y,

0102xycos(x31)dydx=01cos(x31)[(2x)220]dx=012x2cos(x31)dx

Compute the integral with respect to x

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