Chapter 15.4, Problem 8E

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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.*

**8.** *D* is bounded by *y* = *x* + 2 and *y* = *x*^{2}; *ρ*(*x*, *y*) = *kx*^{2}

To determine

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

Explanation

**Given:**

The region *D* is bounded by

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:**

Solve the given equations and obtain the value of *x*.

That is, *x* varies from

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

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

=

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* and apply the corresponding limit.

Further simplify the terms as shown below

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