   Chapter 15.2, Problem 64E

Chapter
Section
Textbook Problem

In evaluating a double integral over a region D, a sum of iterated integrals was obtained as follows: ∬ D f ( x ,   y )   d A   =   ∫ 0 1 ∫ 0 2 y f ( x ,   y )   d x   d y + ∫ 1 3 ∫ 0 3 − y f ( x ,   y )   d x   d y Sketch the region D and express the double integral as an iterated integral with reversed order of integration.

To determine

To reverse: The order of integration and sketch the outline of the region D.

Explanation

Given:

0x2y,0y1 and 0x3y,1y3 .

It is given that whenever y lies between 0 and 1,x varies from 0 to 2y. Similarly, whenever y varies from 1 to 3, x varies from 0 to 3y . Graph the given information as given below in the Figure 1.

From the Figure 1, it is observed that x varies from 0 to 2 and y varies from x2 to 3x . Therefore, the iterated integral becomes,

Df(x,y)dA=0</

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