Chapter 15.4, Problem 13E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

The boundary of a lamina consists of the semicircles
*x*-axis that join them. Find the center of mass of the lamina if the density at any point is proportional to its distance from the origin.

To determine

**To find:** The center of mass of the lamina occupied by the given disk *D*.

Explanation

**Given:**

The region *D* is the semi annular region of the covered by

The density function is proportional to the distance from the origin, that is

**Formula used:**

The total mass of the lamina is,

Here, the density function is given by
*D* is the region that is occupied by the lamina.

The center of mass of the lamina that occupies the given region *D* is

Here,

If
*R* given by

If

**Calculation:**

Convert into polar coordinates to make the problem easier. So, from the given conditions it is observed that *r* varies from 1 to 2 and

Integrate with respect to *r* and

=

In order to get the coordinates of the center of mass, find

Integrate with respect to *r* and

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