Chapter 15.8, Problem 38E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Use cylindrical or spherical coordinates, whichever seems more appropriate.

**38**. A solid right circular cone with constant density has base radius *a* and height *h*.

(a) Find the moment of inertia of the cylinder about its axis.

(b) Find the moment of inertia of the cylinder about a diameter of its base.

(a)

To determine

**To find:** The moment of inertia about its axis for the given region by using spherical or cylindrical coordinates.

Explanation

**Given:**

The region *B* is the right circular cone with radius *a* and height *h.*

The density function is a constant *K.*

**Formula used:**

If
*E* given by

If
*z*, then,

The cylindrical coordinates

**Calculation:**

The base of the right circular cone lies in the *xy-*plane. So, its axis is *z-*axis. Here, it is more appropriate to use the cylindrical coordinates because the given solid is a cone. Thus, the moment of inertia about *z* axis is given by,

Integrate with respect to *z* and apply the limit

(b)

To determine

**To find:** The moment of inertia about the diameter of its base for the region in part (a).

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