Define product of inertia

The product of inertia of an area about a pair of perpendicular axes $x$ and $y$ is the sum, over the whole area, of the products of each elemental area and its two coordinates:

$$I_{xy} = \int xy\,dA$$

Its unit is length$^4$ (for example mm$^4$). Unlike $I_x$ and $I_y$, it may be positive, negative or zero: it is zero if either axis is an axis of symmetry, and it is positive when most of the area lies in the first and third quadrants. By the parallel-axis theorem $I_{xy} = \bar I_{xy} + A\bar x\bar y$.