Chapter 15.7, Problem 26E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Use cylindrical coordinates.

26. (a) Find the volume of the solid that the cylinder *r* = *a* cos
*a* centered at the origin.

(b) Illustrate the solid of part (a) by graphing the sphere and the cylinder on the same screen.

(a)

To determine

**To find:** The volume of the given solid by using cylindrical coordinates.

Explanation

**Given:**

The region *E* is the cylinder
*a* centered at origin.

**Formula used:**

If
*E* given by

The cylindrical coordinates

**Calculation:**

Equation of the sphere of radius *a* is

Thus, the value of *z* is obtained below.

Therefore, *r* varies from 0 to
*z* varies from

Use the formula mentioned above to change the given problem into cylindrical coordinates. Then, by the equation (1), the volume of the given solid is computed as follows.

Integrate with respect to *z* and apply the limit of it.

Integrate with respect to *r* and apply the limit of it. For that substitute

On further simplification, the required integral value is obtained

(b)

To determine

**To illustrate:** The given solid mentioned in part (a) by using graphing calculator.

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