Mass Moment of Inertia

The mass moment of inertia $I$ about a specified reference axis is defined as:

$$I=\int r^{2}dm$$

where $r$ is the distance from the reference axis to the mass element $dm$.

Parallel-Axis Theorem

If the rotation axis of a rigid body does not coincide with the body’s axis of symmetry, but is parallel to it at a distance $d$, then the mass moment of inertia about the given rotation axis is given by:

$$I=I_s+m{d}^2$$

where $I_s$ is the inertia about the symmetry axis.