   Chapter 15.6, Problem 35E

Chapter
Section
Textbook Problem

Write five other iterated integrals that are equal to the given iterated integral.35. ∫ 0 1 ∫ y 1 ∫ 0 y   f ( x ,   y ,   z )   d z   d x   d y

To determine

To express: The integral 01y10yf(x,y,z)dzdxdy in five different ways.

Explanation

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 1 and y varies from 0 to x and z varies from 0 to y.

Hence, E={(x,y,z)|0x1,0yx,0zy}

Therefore, 010x0yf(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 1 and z varies from 0 to y and x varies from y to 1.

Hence, E={(x,y,z)|0y1,0zy,yx1}

Therefore, 010yy1f(x,y,z)dxdzdy

Also, from Figure 2, it is observed that z varies from 0 to 1 and y varies from z to 1 and x varies from y to 1.

Hence, E={(x,y,z)|0z1,zy1,yx1}

Therefore, 01z1y1f(x,y,z)dxdydz

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 1 and z varies from 0 to x and y varies from z to x.

Hence, E={(x,y,z)|0x1,0zx,zyx}

Therefore, 010xzxf(x,y,z)dydzdx

Also, from Figure 3, it is observed that z varies from 0 to 1 and x varies from z to 1 and y varies from z to x

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