Chapter 15.4, Problem 24E

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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

**24**. The region under the curve *y* = sin *x* from *x =* 0 to *x =*

To determine

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

Explanation

**Given:**

The density function,

The region *D* is under the curve

**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,

**Calculation:**

From the given conditions, it is observed that, *x* varies from 0 to
*y* varies from 0 to

Obtain the moment of inertia

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

Integrate with respect to *x.* For that substitute

On further simplification, the value of

Obtain the moment of inertia

Integrate with respect to *y*

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