   Chapter 15.2, Problem 53E

Chapter
Section
Textbook Problem

Evaluate the integral by reversing the order of integration. ∫ 0 1 ∫ x 1 y 3   + 1   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)=y3+1 .

The domain D is, D={(x,y)|0x1,xy1} .

Calculation:

b

Reverse the order of integration, D will become D={(x,y)|0xy2,0y1} . The value of the double integral is,

Df(x,y)dA=01x1y3+1dydx=010y2y3+1dxdy

First, compute the integral with respect to x.

010y2y3+1dxdy=01[0y2y3+1dx]dy=01y3+1[x]0y2dy

Apply the limit value for x,

010y2y3+1dxdy=01y3+1[y20]dy=01y2y3+1dy

Compute the integral with respect to y

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