   Chapter 15.2, Problem 32E

Chapter
Section
Textbook Problem

Find the volume of the given solid.32. Bounded by the cylinders x2 + y2 = r2 and y2 + z2 = r2

To determine

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

Explanation

Given:

The function is y2+z2=r2 .

The cylinder is, x2+y2=r2 .

Formula used:

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

Calculation:

The given function can be expressed as z=r2y2 . It is observed from the given equations of cylinders that y varies from −r and r and x varies from r2y2 to r2y2 . Thus, the volume of the solid is computed as follows.

V=RzdA=rrr2y2r2y2r2y2dxdy

First, compute the integral with respect to x.

V=rr[xr2y2]r2y2r2y2dy

Apply the limit value for x,

V=rr[((r2y2)r2y2)((<

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