(i) Moment of inertia of a hollow sphere about a diameter:

Let us consider a hollow sphere of inner radius $r_1$ and outer radius $r_2$. So the volume of this hollow sphere is $\frac{4}{3}\pi(r_2^3-r_1^3)$. If M be the mass of the hollow sphere, then mass density i.e., mass per unit volume of the hollow sphere is $\frac{M}{\frac{4}{3}\pi(r_2^3-r_1^3)}$ = $\frac{3M}{4\pi(r_2^3-r_1^3)}$.

In order to calculate the moment of inertia of this hollow sphere about the diameter CD, we can assume two concentric spheres, one of radius $r_1$ and the other is of radius $r_2$. The mass of the sphere of radius $r_1$ is $M_1=\frac{4}{3}\pi{r_1}^3\times{\frac{3M}{4\pi(r_2^3-r_1^3)}}$ = $\frac{M{r_1}^3}{(r_2^3-r_1^3)}$. The mass of the sphere of radius $r_2$ is $M_2=\frac{4}{3}\pi{r_2}^3\times{\frac{3M}{4\pi(r_2^3-r_1^3)}}$ = $\frac{M{r_2}^3}{(r_2^3-r_1^3)}$.

The moment of inertia of the sphere of radius $r_1$ about the diameter CD is
$I_1=\frac{2}{5}M_1{r_1}^2\\=\frac{2}{5}\frac{M{r_1}^3}{(r_2^3-r_1^3)}{r_1}^2\\=\frac{2}{5}\frac{M{r_1}^5}{(r_2^3-r_1^3)}$

[ To know the derivation of the moment of inertia of a solid sphere about its diameter Click Here.]

The moment of inertia of the sphere of radius $r_2$ about the diameter CD is,

$I_2=\frac{2}{5}M_2{r_2}^2\\=\frac{2}{5}\frac{M{r_2}^3}{(r_2^3-r_1^3)}{r_2}^2\\=\frac{2}{5}\frac{M{r_2}^5}{(r_2^3-r_1^3)}$

So the moment of inertia of the hollow sphere about the diameter CD is ,
$\displaystyle{I=I_2-I_1\\or,\ I=\frac{2}{5}\frac{(r_2^5-r_1^5)}{(r_2^3-r_1^3)}}$

(ii) Moment of inertia of a hollow sphere about its tangent:

By applying theorem of parallel axes, we can calculate the moment of inertia of the hollow sphere about a tangent EF, which is parallel to the axis CD, and the perpendicular distance between these two axes CD and EF is $r_2$.

Therefore the moment of inertia of the hollow sphere about the axis EF is

$I_{EF}=I+Mr_2^2$

or, $\displaystyle{I_{EF}=\frac{2}{5}\frac{(r_2^5-r_1^5)}{(r_2^3-r_1^3)}+Mr_2^2}$

(iii) The moment of inertia of the hollow sphere about its centre:

Let us consider a hollow sphere of inner radius $r_1$ and outer radius $r_2$.So the volume of the sphere is $\frac{4}{3}\pi(r_2^3-r_1^3)$. If M be the mass of this hollow sphere, then the mass density i.e., mass per unit volume of this hollow sphere is $\frac{M}{\frac{4}{3}\pi(r_2^3-r_1^3)}$ = $\frac{3M}{4\pi(r_2^3-r_1^3)}$.

In order to calculate the moment of inertia of this hollow sphere about the centre O, let us consider an elementary concentric spherical shell of radius $x$ and thickness $dx$ with centre at O.

So the volume of this shell is $4\pi{x^2}dx$ and the mass is $4\pi{x^2}dx\times{\frac{3M}{4\pi(r_2^3-r_1^3)}}$ = $\frac{3M}{r_2^3-r_1^3}x^2{dx}$.

The moment of inertia of this elementary shell about the centre O is $\frac{3M}{r_2^3-r_1^3}x^2{dx}\times{x^2}$ = $\frac{3M}{r_2^3-r_1^3}x^4{dx}$

Now the moment of inertia of the whole hollow sphere about the centre $O$ is given by,

$I_0=\displaystyle{\int_{r_1}^{r_2}}\frac{3M}{r_2^3-r_1^3}x^4{dx}\\=\frac{3M}{5(r_2^3-r_1^3)}{\left(x^5\right)}_{r_1}^{r_2}$

or, $\displaystyle{I_0=\frac{3}{5}M\frac{(r_2^5-r_1^5)}{(r_2^3-r_1^3)}}$