Parallel-Axis Theorem Let \(\mathcal{W}\) be a region in \(\mathbf{R}^{3}\) with center of mass at the origin. Let \(I_{z}\) be the moment of inertia of \(\mathcal{W}\) about the \(z\)-axis, and let \(I_{h}\) be the moment of inertia about the vertical axis through a point \(P=(a, b, 0)\), where \(h=\sqrt{a^{2}+b^{2}}\). By definition,

\[I_{h}=\iiint_{\mathcal{W}}\left((x-a)^{2}+(y-b)^{2}ight) \delta(x, y, z) d V\]

Prove the Parallel-Axis Theorem: \(I_{h}=I_{z}+M h^{2}\).