Area Moment of Inertia

The area moment of inertia is the second moment of an area around a given axis:

$$I_{A,\underline{x}} = \int_A r_{\underline{x}}^2 dA$$

where $A$ is the total area, $dA$ denotes a differential element of the area, and $r_{\underline{x}}$ denotes the distance of the differential element from the axis of rotation $\underline{x}$.

Comparing with the definition of mass moment of inertia in §F.4.3 above, we see that mass is replaced by area in the area moment of inertia.

In a planar mass distribution with total mass $M$ uniformly distributed over an area $A$ (i.e., a constant mass density of $\rho=M/A$), the mass moment of inertia $I_\rho$ is given by the area moment of inertia times mass-density $\rho$:

$$I_\rho \isdef \int_M r^2 dM = \int_A r^2 \rho\, dA = \rho I_A$$