Consider a steel plate of arbitrary shape. The calculation of the product of inertia for the plate area involves using a Cartesian coordinate system and choosing a differential area element on the plate. The product of the differential area coordinates multiplied with the area when integrated over the entire region gives the product of inertia. This value can be positive or negative depending on the sign of the coordinates, and zero if either axis is part of the area's symmetry. Rewriting the coordinates with respect to the centroidal axes in the product of inertia expression and simplifying yields four terms. The first term represents the product of inertia about the centroidal axes. The following two integrals are the moments of the area about the centroidal axis, and so, reduce to zero. The fourth integral term gives the total area. The final expression obtained is the parallel- axis theorem for the product of inertia. The product of inertia is helpful in determining the maximum and minimum moments of inertia for any area.