Chapter 16.2, Problem 37E

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8th Edition

James Stewart

ISBN: 9781285741550

Textbook Problem

If a wire with linear density *ρ*(*x*, *y*) lies along a plane curve *C*, its **moments of inertia** about the *x*- and *y*-axes are defined as

*I _{x} =* ∫

Find the moments of inertia for the wire in Example 3.

To determine

**To find:** The moment of inertia for a wire about the *x*-axis, and *y*-axis.

Explanation

**Given data:**

The density

The moment of inertia of the wire about *x*-axis is defined as follows.

The moment of inertia of the wire about *y*-axis is defined as follows.

**Formula used:**

Write the expression for density function as follows.

Here,

Write the expression for

Consider the parametric equations of the plane curve

**Calculation of density function:**

Substitute

**Calculation of**
**:**

Substitute

**Calculation of moment of inertia about x-axis**

Substitute

Simplify the expression as follows.

Simplify the expression as follows.

Apply substitution method.

Find new limits

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