91影视

The value of the sphere is as follows:

${\mathrm{x}}^2+{\mathrm{y}}^2+{\mathrm{z}}^2=4$

The area of-coordinates of the centroid using the total moments in the-direction is given below:

$\overline{\mathbf{x}}\mathbf{=}\frac{\mathbf{total}\mathbf{}\mathbf{moments}}{\mathbf{total}\mathbf{}\mathbf{area}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{A}}{\mathbf{鈭�}}_{\mathbf{c}}^{\mathbf{d}}\mathbf{xf}\mathbf{\left(}\mathbf{x}\mathbf{\right)}\mathbf{dx}$

And, considering the moments in the -direction about the -axis and re-expressing the function in terms of is given below:

$\overline{\mathbf{y}}\mathbf{=}\frac{\mathbf{total}\mathbf{}\mathbf{moments}}{\mathbf{total}\mathbf{}\mathbf{area}}\mathbf{=}\frac{\mathbf{1}}{\mathbf{A}}{\mathbf{鈭�}}_{\mathbf{c}}^{\mathbf{d}}\mathit{y}\mathbf{f}\mathbf{\left(}\mathit{y}\mathbf{\right)}\mathbf{dy}$

The volume is as follows:

$$\begin{gathered} \mathrm{V}={鈭�}_0^{2\mathrm{蟺}}\mathrm{诲蠁}{鈭�}_0^{\mathrm{a}}\mathrm{蝉颈苍胃诲胃}{鈭�}_{\frac{1}{\mathrm{肠辞蝉胃}}}^2{\mathrm{r}}^2\mathrm{dr} \\ =2\underset{}{鈭�}{鈭�}_0^{\frac{\underset{}{鈭�}}{3}}\mathrm{蝉颈苍胃诲胃}{\overline{)\frac{1}{3}{\mathrm{r}}^3}}^2\frac{1}{\mathrm{肠辞蝉胃}} \\ =\frac{2\underset{}{鈭�}}{3}{鈭�}_0^{\frac{\underset{}{鈭�}}{3}}\mathrm{蝉颈苍胃诲胃}\left(8-\frac{1}{{\cos }^3\mathrm{胃}}\right) \\ =\frac{2\underset{}{鈭�}}{3}\left[\overline{)-8\mathrm{肠辞蝉胃}}\overline{){\frac{\mathrm{蟺}}{3}}_0}+{鈭�}_0^{\frac{\mathrm{蟺}}{3}}\frac{\mathrm{d}\left(\mathrm{肠辞蝉胃}\right)}{{\cos }^3\mathrm{胃}}\right] \\ =\frac{2\underset{}{鈭�}}{3}\left[4-\frac{1}{2}\frac{1}{{\cos }^2\mathrm{胃}}\overline{)0^{\frac{\mathrm{蟺}}{3}}}\right] \\ =\frac{5\mathrm{蟿蟿}}{3} \\ \mathrm{Hence},\mathrm{the}\mathrm{volume}\mathrm{is}\mathrm{V}=\frac{5\mathrm{蟿蟿}}{3} \end{gathered}$$

The coordinates of the centroid is as follows:

$$\begin{gathered} \mathrm{V}\overline{\mathrm{z}}={鈭�}_0^{2\mathrm{蟿蟿}}\mathrm{诲胃}{鈭�}_1^2\mathrm{zdz}{鈭�}_0^{\sqrt{4-{\mathrm{z}}^2}}\mathrm{rdr}=2\mathrm{蟿蟿}{鈭�}_1^2\mathrm{zdz}\frac{1}{2}{\mathrm{r}}^2\overline{)0^{\sqrt{4-{\mathrm{z}}^2}}} \\ =\mathbf{蟿蟿}{鈭�}_1^2\mathrm{zdz}\left(4-{\mathrm{z}}^2\right)=\mathrm{蟿蟿}\left[2{\mathrm{z}}^2-\frac{1}{4}{\mathrm{z}}^4\right]\overline{)1^2=\frac{9\mathrm{蟿蟿}}{4}} \end{gathered}$$

The z-coordinate of the centroid is as follows:

$\overline{\mathrm{z}}=\frac{9\mathrm{蟺}}{4\mathrm{V}}=\frac{27}{20}$

Hence, the z-coordinate of the centroid $\overline{\mathrm{z}}=\frac{27}{20}$.

$$\begin{gathered} \mathrm{V}={鈭�}_0^{2\mathrm{蟿蟿}}\mathrm{诲蠁}{鈭�}_0^{\mathrm{a}}\mathrm{蝉颈苍胃诲胃}{鈭�}_{\frac{1}{\mathrm{肠辞蝉胃}}}^2{\mathrm{r}}^2\mathrm{dr} \\ =2\mathrm{蟿蟿}{鈭�}_0^{\frac{\mathrm{蟿蟿}}{3}}\mathrm{蝉颈苍胃诲胃}\frac{1}{3}{\mathrm{r}}^3{{\overline{)}}^2}_{\frac{1}{\mathrm{肠辞蝉胃}}} \\ =\frac{2\mathrm{蟿蟿}}{3}{鈭�}_0^{\frac{\mathrm{蟿蟿}}{3}}\mathrm{蝉颈苍胃诲胃}\left(8-\frac{1}{{\cos }^3\mathrm{胃}}\right) \\ =\frac{2\mathrm{蟿蟿}}{3}\left[-8\mathrm{肠辞蝉胃}\overline{){\frac{\mathrm{蟿蟿}}{3}}_0+{鈭�}_0^{\frac{\mathrm{蟿蟿}}{3}}}\frac{\mathrm{d}\left(\mathrm{肠辞蝉胃}\right)}{{\cos }^3\mathrm{胃}}\right] \\ =\frac{2\mathrm{蟿蟿}}{3}\left[4-\frac{1}{2}\frac{1}{{\cos }^2\mathrm{胃}}\overline{)0^{\frac{\mathrm{蟿蟿}}{3}}}\right] \\ =\frac{5\mathrm{蟿蟿}}{3} \\ \mathrm{V}\overline{\mathrm{z}}={鈭�}_0^{2\mathrm{蟿蟿}}\mathrm{诲胃}{鈭�}_1^2\mathrm{zdz}{鈭�}_0^{\sqrt{4-{\mathrm{z}}^2}}\mathrm{rdr}=2\mathrm{蟿蟿}{鈭�}_1^2\mathrm{zdz}\frac{1}{2}{\mathrm{r}}^2\overline{)0^{\sqrt{4-{\mathrm{z}}^2}}} \\ =\mathrm{蟿蟿}{鈭�}_1^2\mathrm{zdz}\left(4-{\mathrm{z}}^2\right)=\mathrm{蟿蟿}\left[2{\mathrm{z}}^2-\frac{1}{4}{\mathrm{z}}^4\right]\overline{)1^2=\frac{9\mathrm{蟿蟿}}{4}} \end{gathered}$$