Chapter 15, Problem 41RE

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

Consider a lamina that occupies the region *D* bounded by the parabola *x =* 1 − *y*^{2} and the coordinate axes in the first quadrant with density function *ρ*(*x*, *y*) = *y*.

(a) Find the mass of the lamina.

(b) Find the center of mass.

(c) Find the moments of inertia and radii of gyration about the *x-* and *y*-axes.

(a)

To determine

**To find:** The 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:**

From the given conditions, it is observed that *x* varies from 0 to 1. Then, the mass of the lamina is,

(b)

To determine

**To find:** The center of the mass.

(c)

To determine

**To find:** The moments of inertia and radius of gyration about *x* and *y* axis.

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