91Ó°ÊÓ

The ball $r≤a$is cut by the cone $\mathrm{θ}=\mathrm{Î\pm }≤\mathrm{Ï„}\mathrm{Ï„}/2$.

The spherical coordinates are related to Cartesian coordinates $\left(x,y,z\right)$ by:

role="math" localid="1659167216419" $\begin{array}{l}\mathbf{r}\mathbf{=}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}\mathbf{+}{\mathbf{z}}^{\mathbf{2}}}\mathbf{θ}\\ \mathbf{=}\mathbf{t}\mathbf{a}{\mathbf{n}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\frac{\mathbf{x}}{\mathbf{y}}\mathbf{\right)}\mathbf{ϕ}\\ \mathbf{=}\mathbf{c}\mathbf{o}{\mathbf{s}}^{\mathbf{-}\mathbf{1}}\mathbf{\left(}\frac{\mathbf{z}}{\mathbf{r}}\mathbf{\right)}\end{array}$

The cylindrical coordinates are related to Cartesian coordinates$\mathbf{(}\mathbf{x}\mathbf{,}\mathbf{y}\mathbf{,}\mathbf{z}\mathbf{)}$by:

$\begin{array}{l}\mathbf{r}\mathbf{=}\sqrt{{\mathbf{x}}^{\mathbf{2}}\mathbf{+}{\mathbf{y}}^{\mathbf{2}}}\mathbf{θ}\\ \mathbf{=}\mathbf{t}\mathbf{a}{\mathbf{n}}^{\mathbf{-}\mathbf{1}}\left(\frac{y}{x}\right)\mathbf{z}\\ \mathbf{=}\mathbf{z}\end{array}$

(a)

Calculate the required volume as follows:

$$\begin{gathered} V={∫}_0^a{r}^{2}dr{∫}_0^a\sin \mathrm{θ}d\mathrm{θ}{∫}_0^{2\mathrm{Ï„}\mathrm{Ï„}}d\mathrm{Ï•} \\ =\frac{2\mathrm{Ï„}\mathrm{Ï„}{a}^3}{3}{\left[-\cos \mathrm{θ}\right]}_0^{\mathrm{Î\pm }} \\ =\frac{2\mathrm{Ï„}\mathrm{Ï„}{a}^3}{3}\left(1-\cos \mathrm{Î\pm }\right) \end{gathered}$$

Thus, the required volume is $\frac{2\mathrm{Ï„}\mathrm{Ï„}{a}^3}{3}(1-cos\mathrm{Î\pm })$.

(b)

Substituting the value and calculating as follows:

$$\begin{gathered} \overline{z}=\frac{1}{V}{∫}_0^a{∫}_0^{2\mathrm{Ï„}\mathrm{Ï„}}{∫}_0^{\mathrm{Î\pm }}z{r}^{2}dr\sin \mathrm{θ}d\mathrm{θ}d\mathrm{Ï•} \\ =\frac{2\mathrm{Ï„}\mathrm{Ï„}{a}^3}{V}{∫}_0^a{r}^{3}dr{∫}_0^{\mathrm{Î\pm }}\sin \mathrm{θ}\cos \mathrm{θ}\mathrm{θ}d\mathrm{θ} \\ =\frac{\mathrm{Ï„}\mathrm{Ï„}{a}^4}{8V}{\left[-\cos \left(2\mathrm{θ}\right)\right]}_0^{\mathrm{Î\pm }} \\ =\frac{\mathrm{Ï„}\mathrm{Ï„}{a}^4}{8V}\left(1-\cos \mathrm{Î\pm }\left(2\mathrm{Î\pm }\right)\right)........\left(1\right) \end{gathered}$$

Substitute $\frac{2\mathrm{Ï„}\mathrm{Ï„}{a}^3}{3}(1-cos\mathrm{Î\pm })$for V in Equation (1) as follows:

$$\begin{gathered} \overline{z}=\frac{\mathrm{Ï„}\mathrm{Ï„}{a}^4}{8}\left(1-\cos \left(2\mathrm{Î\pm }\right)\right)\left(\frac{3}{2\mathrm{Ï„}\mathrm{Ï„}{a}^3\left(1-\cos \mathrm{Î\pm }\right)}\right) \\ =\frac{3a}{16}\frac{{\sin }^2\mathrm{Î\pm }}{{\sin }^2\left(\mathrm{Î\pm }/2\right)} \\ =\frac{3a}{4}{\cos }^2\left(\mathrm{Î\pm }/2\right) \\ =\frac{3a}{8}\left(1+\cos \mathrm{Î\pm }\right) \end{gathered}$$