Chapter 15.4, Problem 12E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Find the center of mass of the lamina in Exercise 11 if the density at any point is proportional to the square of 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 disk

The density function is proportional to the square of its 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 0 to 1 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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