Find the moment of inertia of a circle of radius R

Find the moment of inertia of a circle of radius $R$ with respect to one of its diameters.

Step 1: Set up in polar coordinates

\[I_x=\int_0^{2\pi}\int_0^R(\rho\sin\theta)^2\rho\,d\rho\,d\theta=\left(\int_0^{2\pi}\sin^2\theta\,d\theta\right)\left(\int_0^R\rho^3\,d\rho\right)\]

Step 2: Evaluate each factor

\[\int_0^{2\pi}\sin^2\theta\,d\theta=\pi,\qquad \int_0^R\rho^3\,d\rho=\frac{R^4}4\]

Step 3: Combine

\[I_x=\pi\cdot\frac{R^4}4=\frac{\pi R^4}4\]

\[\boxed{I_x=\frac{\pi R^4}4}\]

By symmetry, $I_y=\dfrac{\pi R^4}4$ as well, and with $A=\pi R^2$ this is $\dfrac{AR^2}4$; the radius of gyration is $k=\dfrac R2$.