   Chapter 15, Problem 53RE

Chapter
Section
Textbook Problem

Rewrite the integral ∫ − 1 1 ∫ x 2 1 ∫ 0 1 − y f ( x , y , z )   d z   d y   d x as an iterated integral in the order dx dy dz.

To determine

To rewrite: The given integral in a mentioned order.

Explanation

Given:

The iterated integral, 11x2101yf(x,y,z)dzdydx.

Calculation:

Since in the required order dx is coming first, it is necessary to project the graph of the function into the yz-plane. The equation of x becomes,

x2=yx=±y

And

z=1yy=1z

Since the projection is in the yz-plane, substitute x=0. And equate the above equations as given below,

z=1yz=10          [y=x=0]z=1

Thus, from the above equations, it is observed that x varies from y to y, y varies from 0 to 1z and z varies from 0 to 1

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