   Chapter 15.2, Problem 29E

Chapter
Section
Textbook Problem

Find the volume of the given solid.29. Enclosed by the cylinders z = x2, y = x2 and the planes z = 0, y = 4

To determine

To find: The volume of the solid that is enclosed by the cylinders and planes.

Explanation

Given:

The function is z=x2 .

The planes are, y=4,z=0 .

The cylinder is y=x2 .

Formula used:

The volume of the solid, V=DzdA , where, z is the given function.

Calculation:

It is observed from the given equations of planes and cylinders that y varies from x2 and 4 and x varies from 2 to 2. Thus, the volume of the solid is computed as follows.

V=RzdA=22x24x2dydx

First, compute the integral with respect to y.

V=22[x2y]x24dx

Apply the limit value for y,

V=22[(x2(4)x2(x2))]dx=22[4x2x4]dx

Compute the integral with respect to x

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