Second moment of area

Remarks

Example: for the rectangle $\abs{y} \le h/2$, $\abs{z} \le b/2$, the moment about the centroidal $z$-axis is $I_z = \int_{-h/2}^{h/2} y^2 b \, dy = bh^3/12$. For a disc of radius $R$ centred at the origin, $I_y = I_z = \pi R^4/4$ and $J = \pi R^4/2$.

Non-example: the integral $\int_A y \, dA$ is the first moment of area; it is $0$ about a centroidal axis, so it does not measure resistance to bending.

Let $I_c$ be the moment about a centroidal axis and $\abs{A}$ the area. The parallel axis theorem gives $I = I_c + d^2 \abs{A}$ for a parallel axis at distance $d$. The symbol $I$ is used both for this quantity and for the mass moment of inertia, which has the mass density in the integral.