Chapter 15.4, Problem 15E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Find the center of mass of a lamina in the shape of an isosceles right triangle with equal sides of length *a* if the density at any point is proportional to the square of the distance from the vertex opposite the hypotenuse.

To determine

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

Explanation

**Given:**

The region *D* is the isosceles triangle with equal sides of length *a*.

The density function is proportional to the square of the distance from the vertex opposite to the hypotenuse, 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,

**Calculation:**

From the given conditions it is observed that *x* varies from 0 to *a* and *y* varies from 0 to

Integrate with respect to *y* and apply the corresponding limit.

=

Integrate with respect to *x* using the technique of integration by parts.

Let

Then,

Thus, the total mass of the lamina is obtained as follows.

Apply the limit of *x*.

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

Integrate with respect to *y* and apply the corresponding limit.

Further simplify the terms as shown below

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