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Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550

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BuyFindarrow_forward

Calculus: Early Transcendentals

8th Edition
James Stewart
ISBN: 9781285741550
Textbook Problem

The figure shows the region of integration for the integral 0 1 0 1 x 2 0 1 x f ( x , y , z ) d y d z d x

Rewrite this integral as an equivalent iterated integral in the five other orders.

images

To determine

To express: The integral 0101x201xf(x,y,z)dydzdx in five different ways.

Explanation

Given:

The region of the given integral is given in the form of figure.

The region are y=1x and z=1x2.

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 xz-plane.

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

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

Therefore, 0101z01xf(x,y,z)dydxdz

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

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

From Figure 2, it is observed that y varies from 0 to 1, x varies from 0 to 1y and z varies from 0 to 1x2

Therefore, 0101y01x2f(x,y,z)dzdxdy

Also, from Figure 2, observed that x varies from 0 to 1, y varies from 0 to 1x and z varies from 0 to 1x2

0101x01x2f(x,y,z)dzdydx

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

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

In this plane, the surface z=1x2 intersects the plane y=1x. Therefore equate both the planes to get z=2yy2 so, the projection on this plane is divided into two regions named as R1 and R2. In this region R1, x varies from 0 to 1y and R2, x varies from 0 to 1z. So, the integral is equal.

The integral for R1 is, 01011z01zf(x,y,z)dxdydz+0111z101yf(x,y,z)dxdydz

The integral for R1 is, 0102yy201yf(x,y,z)dxdzdy+012yy2101zf(x,y,z)dxdzdy

Hence, the iterated integral 0101x201xf(x,y,z)dydzdx is expressed in 5 different ways as follows

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Chapter 15 Solutions

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Sect-15.1 P-11ESect-15.1 P-12ESect-15.1 P-13ESect-15.1 P-14ESect-15.1 P-15ESect-15.1 P-16ESect-15.1 P-17ESect-15.1 P-18ESect-15.1 P-19ESect-15.1 P-20ESect-15.1 P-21ESect-15.1 P-22ESect-15.1 P-23ESect-15.1 P-24ESect-15.1 P-25ESect-15.1 P-26ESect-15.1 P-27ESect-15.1 P-28ESect-15.1 P-29ESect-15.1 P-30ESect-15.1 P-31ESect-15.1 P-32ESect-15.1 P-33ESect-15.1 P-34ESect-15.1 P-35ESect-15.1 P-36ESect-15.1 P-37ESect-15.1 P-38ESect-15.1 P-39ESect-15.1 P-40ESect-15.1 P-41ESect-15.1 P-42ESect-15.1 P-43ESect-15.1 P-44ESect-15.1 P-47ESect-15.1 P-48ESect-15.1 P-49ESect-15.1 P-50ESect-15.1 P-52ESect-15.2 P-1ESect-15.2 P-2ESect-15.2 P-3ESect-15.2 P-4ESect-15.2 P-5ESect-15.2 P-6ESect-15.2 P-7ESect-15.2 P-8ESect-15.2 P-9ESect-15.2 P-10ESect-15.2 P-11ESect-15.2 P-12ESect-15.2 P-13ESect-15.2 P-14ESect-15.2 P-15ESect-15.2 P-16ESect-15.2 P-17ESect-15.2 P-18ESect-15.2 P-19ESect-15.2 P-20ESect-15.2 P-21ESect-15.2 P-22ESect-15.2 P-23ESect-15.2 P-24ESect-15.2 P-25ESect-15.2 P-26ESect-15.2 P-27ESect-15.2 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P-16ESect-15.9 P-17ESect-15.9 P-18ESect-15.9 P-19ESect-15.9 P-20ESect-15.9 P-21ESect-15.9 P-22ESect-15.9 P-23ESect-15.9 P-24ESect-15.9 P-25ESect-15.9 P-26ESect-15.9 P-27ESect-15.9 P-28ECh-15 P-1RCCCh-15 P-2RCCCh-15 P-3RCCCh-15 P-4RCCCh-15 P-5RCCCh-15 P-6RCCCh-15 P-7RCCCh-15 P-8RCCCh-15 P-9RCCCh-15 P-10RCCCh-15 P-1RQCh-15 P-2RQCh-15 P-3RQCh-15 P-4RQCh-15 P-5RQCh-15 P-6RQCh-15 P-7RQCh-15 P-8RQCh-15 P-9RQCh-15 P-1RECh-15 P-2RECh-15 P-3RECh-15 P-4RECh-15 P-5RECh-15 P-6RECh-15 P-7RECh-15 P-8RECh-15 P-9RECh-15 P-10RECh-15 P-11RECh-15 P-12RECh-15 P-13RECh-15 P-14RECh-15 P-15RECh-15 P-16RECh-15 P-17RECh-15 P-18RECh-15 P-19RECh-15 P-20RECh-15 P-21RECh-15 P-22RECh-15 P-23RECh-15 P-24RECh-15 P-25RECh-15 P-26RECh-15 P-27RECh-15 P-28RECh-15 P-29RECh-15 P-30RECh-15 P-31RECh-15 P-32RECh-15 P-33RECh-15 P-34RECh-15 P-35RECh-15 P-36RECh-15 P-37RECh-15 P-38RECh-15 P-39RECh-15 P-40RECh-15 P-41RECh-15 P-42RECh-15 P-43RECh-15 P-44RECh-15 P-45RECh-15 P-47RECh-15 P-48RECh-15 P-49RECh-15 P-51RECh-15 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