Chapter 15.4, Problem 10E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Find the mass and center of mass of the lamina that occupies the region *D* and has the given density function *ρ.*

**10.** *D* is enclosed by the curves *y* = 0 and *y* = cos *x*, −*π*/2 ≤ *x* ≤ *π*/2; *ρ*(*x*, *y*) = *y*

To determine

**To find:** The total mass and the center of mass of the lamina.

Explanation

**Given:**

The region *D* is bounded by the curves

The density function 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:**

The total mass of the lamina is,

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

Integrate with respect to *x*.

On further simplification, the value of *m* becomes,

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

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

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

Let

Then,

Thus, the value of

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