Moment of inertia

Remarks

For a plane lamina $D \subseteq \mathbb{R}^2$, the moments of inertia about the coordinate axes are $I_x = \iint_D y^2 \delta \, dA$ and $I_y = \iint_D x^2 \delta \, dA$. These are double integrals. For a point mass $m$ at distance $r$ from the axis, the moment of inertia is $m r^2$. Example: a solid cylinder of radius $a$, height $h$ and constant density has mass $M = \delta \pi a^2 h$ and $I_z = \delta \int_0^h \int_0^{2\pi} \int_0^a r^3 \, dr \, d\theta \, dz = \tfrac{1}{2} M a^2$. For rotation about $L$ with angular velocity $\omega$, the kinetic energy is $\tfrac{1}{2} I_L \omega^2$.