Chapter 15.2, Problem 75E

Moment of Inertia Consider a wire of density $\rho (x,y)$ given by the space curve $C:r(t)=x(t)\text{i}+y(t)\text{j}$, $0\le t\le b$.

The moments of inertia about the x- and y-axes are given by

$I_x={\displaystyle {\int }_C{y}^2\rho \left(x,y\right)ds}$ and $I_y={\displaystyle {\int }_C{x}^2}\rho \left(x,y\right)ds$.

In Exercises 75 and 76, find the moments of inertia for the wire of density $\rho$.

A wire along $r(t)=a\cos t\text{i}+a\sin t\text{j}$, where $0\le t\le 2\pi$ and $a>0$, with density $\rho (x,y)=1$.