<Dd> I x y = ∫∫ R y x d x d y (\ displaystyle I_ (xy) = \ iint \ limits _ (R) yx \, \ mathrm (d) x \, \ mathrm (d) y) </Dd> <P> It is sometimes necessary to calculate the second moment of area of a shape with respect to an x ′ (\ displaystyle x') axis different to the centroidal axis of the shape . However, it is often easier to derive the second moment of area with respect to its centroidal axis, x (\ displaystyle x), and use the parallel axis theorem to derive the second moment of area with respect to the x ′ (\ displaystyle x') axis . The parallel axis theorem states </P> <Dl> <Dd> I x ′ = I x + A d 2 (\ displaystyle I_ (x') = I_ (x) + Ad ^ (2)) </Dd> </Dl> <Dd> I x ′ = I x + A d 2 (\ displaystyle I_ (x') = I_ (x) + Ad ^ (2)) </Dd>

Mass moment of inertia vs second moment of area