Prove that moment of inertia of an angular sphere about its diameter . ( Given mass , internal and external radii are M , `R_(1) and R_(2)` respectively ) is
`I = 2/5 M ((R_(2)^(5)-R_(1)^(5))/(R_(2)^(3)-R_(1)^(3)))`

Explanation

To prove the moment of inertia \( I \) of a hollow sphere about its diameter, we will follow a systematic approach. Given that the mass of the sphere is \( M \), and the internal and external radii are \( R_1 \) and \( R_2 \) respectively, we need to show that:

\[ I = \frac{2}{5} M \frac{R_2^5 - R_1^5}{R_2^3 - R_1^3} \]

### Step 1: Define the density of the hollow sphere
The density \( \rho \) of the hollow sphere can be defined as the mass per unit volume. The volume of the hollow sphere is given by the difference in volumes of the outer sphere and the inner sphere:
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