Chapter 15.4, Problem 23E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

A lamina with constant density *ρ* (x, y) *= ρ* occupies the given region. Find the moments of inertia *l _{x}* and

**23**. The part of the disk *x*^{2} *+ y*^{2} ≤ *a*^{2} in the first quadrant

To determine

**To find:** The moments of inertia about *x* and *y* axes

Explanation

**Given:**

The density function,

The region *D* is disk

**Formula used:**

The moments of inertia is,

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 radii of gyration about *y* and *x* axis is respectively

If
*R* given by

If

**Calculation:**

Change the given problem into polar coordinates to make it easier by substituting
*r* varies from 0 to *a* and

Obtain the moments of inertia

Integrate with respect to *r* and

On further simplification, the value of

Obtain the moments of inertia

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