Chapter 15.3, Problem 28E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

(a) A cylindrical drill with radius *r*_{1} is used to bore a hole through the center of a sphere of radius *r*_{2}. Find the volume of the ring-shaped solid that remains.

(b) Express the volume in part (a) in terms of the height *h* of the ring. Notice that the volume depends only on *h*, not on *r*_{1} or *r*_{2}.

(a)

To determine

**To find:** The volume of the given solid.

Explanation

**Given:**

The region *D* lies between the sphere of radius

**Formula used:**

If
*R* given by

If

**Calculation:**

From the given region *D*, let
*r* varies from

Substitute

Integrate the function with respect to *r* and

Let

Then,

Obtain the required volume as follows

(b)

To determine

**To express:** The volume obtained in part (a) in terms of height of the ring.

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