Chapter 15, Problem 42RE

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

A lamina occupies the part of the disk *x*^{2} + *y*^{2} ≤ *a*^{2} that lies in the first quadrant.

(a) Find the centroid of the lamina.

(b) Find the center of mass of the lamina if the density function is *ρ*(*x*, *y*) = *xy*^{2}.

(a)

To determine

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

Explanation

**Given:**

The region *D* is the disk

The density function is proportional to *x-*axis, that is

**Formula used:**

The centroid of the lamina that occupies the given region *D* is

Here, *A* is the area of the given region.

If *R* given by

If

**Calculation:**

Here, the region is the circle of radius *a.* So, use of polar coordinates is a wise choice. Therefore, from the given conditions it is observed that *r* varies from 0 to *a* and

Then, by the equation (1) and (2), the centroid of the lamina is,

(b)

To determine

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

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