91影视

Physics definitions are given.

In a right-handed coordinate system, the unit vectors $\mathit{i}\mathbf{,}\mathit{j}\mathbf{,}\mathit{k}$are vectors of unit magnitude that point in the direction of the $\mathit{x}\mathbf{,}\mathit{y}\mathbf{,}\mathit{z}$axes, respectively. The orthogonal triad of unit vectors, often known as basic vectors, is a set of three unit vectors that are orthogonal to each other.

Define the region under consideration.

$S=\left\{\left(x,y,z\right)鈭�R^{3}X>0,y>>0.z>0and{x}^2+y^2+z^2鈮�1\right\}$

Find the components of inertia.

$$\begin{gathered} l_{xz}=-鈭�\underset{0}{\overset{1}{鈭�}}xzdV \\ =-\underset{0}{\overset{1}{鈭�}}\underset{0}{\overset{\frac{\mathrm{蟺}}{2}}{鈭�}}\underset{0}{\overset{\frac{\mathrm{蟺}}{2}}{鈭�}}r\cos 蠒\sin 胃r^2\sin 胃d蠒d胃dr \\ =-\underset{0}{\overset{1}{鈭�}}{r}^{4}dr{鈭�}_0^{\frac{\mathrm{蟺}}{2}}\cos 蠒d蠒{鈭�}_0^{\frac{\mathrm{蟺}}{2}}{\sin }^2胃d\left(\sin 胃\right) \end{gathered}$$

Continue the evaluation.

$$\begin{gathered} l_{xz}=-\frac{1}{5}路1路\frac{1}{3} \\ =-\frac{1}{15} \end{gathered}$$

Make analysis about integrals of trigonometric ratios.

$$\begin{gathered} \underset{0}{\overset{\frac{\mathrm{蟺}}{2}}{鈭�}}\cos 蠒d蠒=\underset{0}{\overset{\frac{\mathrm{蟺}}{2}}{鈭�}}\sin 蠒d蠒 \\ {l}_{yz}=l_{xz} \\ \underset{0}{\overset{\frac{\mathrm{蟺}}{2}}{鈭�}}{\cos }^2蠒d蠒=\underset{0}{\overset{\frac{\mathrm{蟺}}{2}}{鈭�}}{\sin }^2蠒d蠒 \\ {l}_{yy}=l_{xx} \end{gathered}$$

Apply these equations and evaluate.

$$\begin{gathered} l_{zz}=鈭�鈭�\left(x^2+y^2\right)dV \\ =\underset{0}{\overset{1}{鈭�}}\underset{0}{\overset{\frac{\mathrm{蟺}}{2}}{鈭�}}\underset{0}{\overset{\frac{\mathrm{蟺}}{2}}{鈭�}}{r}^4{\sin }^3胃d蠒d胃dr \\ =\underset{0}{\overset{1}{鈭�r^{4}dr}}\underset{0}{\overset{\frac{\mathrm{蟺}}{2}}{鈭�}}d蠒\underset{0}{\overset{\frac{\mathrm{蟺}}{2}}{鈭�}}{\sin }^3胃d胃 \end{gathered}$$

Continue the evaluation.

Hence the final answer.

$l=\frac{1}{15}\left(\begin{array}{ccc}\mathrm{蟺}& -1& -1\\ -1& \mathrm{蟺}& -1\\ -1& -1& \mathrm{蟺}\end{array}\right)$