Chapter 15.4, Problem 9E

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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.*

**9.** *D* is bounded by the curves *y* = *e*^{−x}, *y* = 0, *x* = 0, *x =* 1; *ρ*(*x, y*) = *xy*

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* by applying the technique of integration by parts.

Let

Then,

Thus, the value of *m* 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.

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

Let

Then,

Thus, the value of

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