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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.

36. The solid enclosed by the parabolic cylinder y = x2 and the planes z = 3y, z = 2 + y

To determine

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

Explanation

Given:

The planes are z=3y,z=2+y .

The parabolic cylinders are, y=x2 .

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=2+y

And, z2=3y .

Two planes intersect at y = 1.

Thus, x varies from 1 to 1 and y varies from x2 to 1 in the interval [1,1] , z1 lies above z2 .

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

V=Dz1dADz2dA=11x21(2+y)dydx11x21(3y)dydx=11x21(2+y3y)dydx=211x21(1y)dydx

First, compute the integral with respect to y

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