Calculate the moment of inertia of
a. a ring of mass `M` and radius `R` about an axis coinciding with the diameter of the ring.
b. as thin disc about an axis coinciding with the diameter.

Explanation

To calculate the moment of inertia for the given objects, we will follow the steps outlined below:

### Part a: Moment of Inertia of a Ring about an Axis Coinciding with the Diameter

1. **Understand the Problem**: We have a ring of mass \( M \) and radius \( R \). We need to find the moment of inertia about an axis that coincides with the diameter of the ring.

2. **Use the Perpendicular Axis Theorem**: The perpendicular axis theorem states that for a planar body, the moment of inertia about an axis perpendicular to the plane (z-axis) is equal to the sum of the moments of inertia about two perpendicular axes (x and y) lying in the plane of the body. Mathematically, this is expressed as:
\[
...