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Circular Disk Rotating in Its Own Plane

For example, the moment of inertia for a uniform circular disk of total mass $ M$ and radius $ R$, rotating in its own plane about a rotation axis piercing its center, is given by

$\displaystyle I = \frac{M}{\pi R^2}\int_{-\pi}^\pi \int_0^R r^2\, r\,dr\,d\thet...
...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.

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