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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

Find the volume of the solid by subtracting two volumes.

35. The solid enclosed by the parabolic cylinders y = 1 − x2, y = x2 − 1 and the planes x + y + z = 2, 2x + 2yz + 10 = 0

To determine

To find: The volume of the solid that lies in between two planes.

Explanation

Given:

The planes are x+y+z=2,2x+2yz+10=0 .

The parabolic cylinders are, y=1x2,y=x21 .

Formula used:

The volume of the solid, V=Dz1dADz2dA , where, z1 and z2 are the given function.

Calculation:

Express the given plane equation as follows:

z1=2x2y10z1=2x+2y+10

And, z2=2xy

The parabolas y=1x2,y=x21 intersect at −1 and 1. Therefore, x varies from −1 to 1 and y varies from y=x21 to y=1x2 in the interval [1,1] , z1 lies above z2 .

Thus, the volume of the solid is computed as follows.

V=Dz1dADz2dA=11x211x2(2x+2y+10)dydx11x211x2(2xy)dydx=11x211x2(2x+2y+102+x+y)dydx=11x211x2(3x+3y+8)dydx

First, compute the integral with respect to y

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