   Chapter 15.6, Problem 36E

Chapter
Section
Textbook Problem

Write five other iterated integrals that are equal to the given iterated integral.36. ∫ 0 1 ∫ y 1 ∫ 0 z   f ( x ,   y ,   z )   d x   d z   d y

To determine

To express: The integral 01y10zf(x,y,z)dxdzdy 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.

The projection on this plane is divided into two regions named as R1 and R2.

Hence, E={(x,y,z)|0y1,yx1,xz1}{(x,y,z)|0y1,0xy,yz1}

Therefore, 01y1x1f(x,y,z)dzdxdy+010yy1f(x,y,z)dzdxdy

Hence, E={(x,y,z)|0x1,0yx,xz1}{(x,y,z)|0x1,xy1,yz1}

Therefore, 010xx1f(x,y,z)dzdydx+01x1y1f(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 z varies from 0 to 1 and y varies from 0 to z and x varies from 0 to z.

E={(x,y,z)|0z1,0yz,0xz}

010z0zf(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 x to 1 and y varies from 0 to z.

E={(x,y,z)|0x1,xz1,0yz}

01x10zf(x,y,z)dydzdx

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

Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

Find the value of k such that (1,k) is equidistant from (0,0) and (2,1).

Finite Mathematics and Applied Calculus (MindTap Course List)

Define a Type I error and a Type Il error and explain the consequences of each.

Statistics for The Behavioral Sciences (MindTap Course List)

In Exercises 19-24, find the functions f + g, f g, fg, and fg. 19. f(x) = x2 + 5; g(x) = x2

Applied Calculus for the Managerial, Life, and Social Sciences: A Brief Approach

True or False: The function whose graph is given at the right is a probability density function.

Study Guide for Stewart's Single Variable Calculus: Early Transcendentals, 8th

In Exercises 116, determine whether the argument is valid. pqqrpr

Finite Mathematics for the Managerial, Life, and Social Sciences 