   Chapter 15, Problem 54RE

Chapter
Section
Textbook Problem

Give five other iterated integrals that are equal to ∫ 0 2 ∫ 0 y 3 ∫ 0 y 2 f ( x , y , z )   d z   d x   d y

To determine

To express: The given integral in five different ways.

Explanation

Given:

The iterated integral is, 020y30y2f(x,y,z)dzdxdy.

Calculation:

Let D1,D2,D3 be the respective projections of E on xy, yz and zx-planes.

The variable D1 is the projection of E on xy-plane.

The graph of the above plane is shown below in Figure 1.

From Figure 1, it is observed that x varies from 0 to 8 and y varies from 0 to x13 and z varies from 0 to y2 and from 0 to x23.

Therefore, 080x130y2f(x,y,z)dzdydx and 080x130x23f(x,y,z)dzdydx.

The variable D2 is the projection of E on yz-plane.

The graph of the above plane is shown below in Figure 2.

From Figure 2, it is observed that y varies from 0 to 2 and z varies from 0 to y2 and x varies from 0 to y3 and from 0 to z32.

Therefore, 020y20y3f(x,y,z)dxdzdy and 020y20z32f(x,y,z)dxdzdy.

The variable D3 is the projection of E on xz-plane.

The graph of the above plane is shown below in Figure 3.

From Figure 3, it is observed that x varies from 0 to 4 and z varies from 0 to x23 and y varies from 0 to z

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