Circular Disk

For example, the moment of inertia for a circular disk of total mass $M$ and radius $R$ rotating about its center is given by

$$I = \frac{M}{\pi R^2}\int_{-\pi}^\pi \int_0^R r^2 r\,dr\,d\theta ... ...c{2M}{R^2}\int_0^R r^3 dr = \frac{2M}{R^2}\frac{1}{4} R^4 = \frac{1}{2} M R^2.$$