Derive an expression for the moment of inertia of a thin spherical shell about a diameter.

Explanation

To derive the expression for the moment of inertia \(I\) of a thin spherical shell about a diameter, we will follow these steps:

### Step 1: Define the Geometry of the Spherical Shell
Consider a thin spherical shell of radius \(r\) and mass \(m\). The shell is hollow, and we want to find the moment of inertia about a diameter.

### Step 2: Consider an Elemental Mass
We will take a small circular strip (element) on the surface of the sphere at an angle \(\theta\) from the vertical axis. The width of this strip is \(d\theta\), and its radius is \(r\).

...