   Chapter 15.2, Problem 30E

Chapter
Section
Textbook Problem

Find the volume of the given solid.30. Bounded by the cylinder y2 + z2 = 4 and the planes x = 2y, x = 0, z = 0 in the first octant

To determine

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

Explanation

Given:

The function is y2+z2=4 .

The planes are, x=2y,x=0,z=0 .

Formula used:

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

Calculation:

The given function can be rewritten as z=4y2 . It is observed from the given equations of planes that y varies from 0 and 2 and x varies from 0 to 2y. Thus, the volume of the solid is computed as follows.

V=RzdA=0202y4y2dxdy

First, compute the integral with respect to x.

V=02[x4y2]02ydy

Apply the limit value for x,

V=02[2y4y20]dy=022y4y2dy

Compute the integral with respect to y

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