Chapter 15, Problem 43RE

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

(a) Find the centroid of a solid right circular cone with height hand base radius *a.* (Place the cone so that its base is in the *xy*-plane with center the origin and its axis along the positive *z*-axis.)

(b) If the cone has density function
*z*-axis).

(a)

To determine

**To find:** The centroid of the given region *E*.

Explanation

**Given:**

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

**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. It is given that the density is constant.

By the symmetric of the given function, it is observed that *x* and *y* coordinates of the centroid is 0. Thus, the third coordinate is given by,

First find the mass of the lamina,

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

Integrate with respect to

(b)

To determine

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

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