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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter 6.3, Problem 38E

a)

To determine

**To find:** the more wood content of the ring from two rings.

Expert Solution

The both napkin rings are having a same quantity of wood.

The volume only depends up on the height of the napkin ring.

Hence, the both napkin rings are having same quantity of wood.

b)

To determine

**To check:** The part (a) answer by computing the volume of a napkin ring using cylindrical shells.

Expert Solution

The volume of a napkin ring is

**Given:**

The height of the napkin ring is *h*.

The radius of the napkin ring is *r*.

The radius of the sphere is *R*.

**Calculation:**

Consider the equation of napkin ring as follows:

The region lies between

Sketch the solid region as shown below in Figure 1.

Refer Figure 1

Calculate the volume using the method of cylindrical shell as follows.

Substitute *r* for *a*, *R* for *b*, and

Consider

Differentiate both sides of the equation.

Calculate the lower limit value of *u* using equation (4).

Substitute *r* for *x* in equation (4).

Calculate the upper limit value of *u* using equation (4).

Substitute *R* for *x* in equation (4).

Apply lower and upper limits for *u* in equation (3).

Substitute *u* for *dx* in equation (3).

Integrate equation (5).

Consider

Substitute

Hence, the volume of a napkin ring is