The connection between second moments and moments of inertia for was developed in Section 6.6 for plane regions. The present section deals with moments of inertia and the associated radii of gyration for three-dimensional regions.

Let $m$ be the total mass of a three-dimensional region $R$ having density $\mathrm{\δ}\left(x\,y\,z\right)$ or $\mathrm{\δ}\left(r\,\mathrm{\θ}\,z\right)$ in Cartesian or cylindrical coordinates, respectively. Table 8.4.1 lists expressions for the moments of inertia (second moments) $I_x\,I_y\,I_z$. For Cartesian coordinates, $\mathrm{dv}$ is one of the six permutations of the differentials $\mathrm{dx}\,\mathrm{dy}\,\mathrm{dz}$. For cylindrical coordinates, $\mathrm{dv}\prime$ is one of the six permutations of the differentials $\mathrm{dz}\,\mathrm{dr}\,d\mathrm{\θ}$.

For $I_x$, the expressions $y^2\+z^2$ and $r^2{\sin }^2\left(\mathrm{\theta }\right)\+z^2$ represent, for a point in $R$, the square of the distance from the $x$-axis.

For $I_y$, the expressions  $x^2\+z^2$ and $r^2{\cos }^2\left(\mathrm{\theta }\right)\+z^2$ represent, for a point in $R$, the square of the distance from the $y$-axis.

For $I_z$, the expressions  $x^2\+y^2$ and $r^2$ represent, for a point in $R$, the square of the distance from the $z$-axis.

Because Maple uses $I$ for the imaginary unit $\sqrt{-1}$, it is troublesome to assign to a symbol such as $I_x$. Hence, whenever such an assignment is needed in the accompanying examples, a symbol such as $\mathrm{Ix}$ will be used instead.

The radii of gyration represent distances from the Cartesian coordinate-axes where, if all the mass $m$ in $R$ were concentrated, the rotational properties of the region would be preserved.

In each of the following examples, the region $R$ and the density $\mathrm{\δ}$ is taken from the corresponding example in Section 8.3.3. For each example, obtain the second moments $I_x\,I_y\,I_z$, and the radii of gyration $k_x\,k_y\,k_z$.