Chapter 15.4, Problem 5E

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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.*

**5.** *D* is the triangular region with vertices (0, 0), (2, 1), (0, 3); *ρ*(*x*, *y*) = *x* + *y*

To determine

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

Explanation

**Given:**

The region *D* is the triangle with vertices

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:**

From the given information, it is clear that one of the sides of the triangle is

The slope of the line

The equation of the line

Similarly, the slope of the line

The equation of the line

Thus, the equations of the sides of the triangle are,

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

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