Find the moment of inertia of a solid cylinder of mass M and radius R about a line parallel to the axis of the cylinder and on the surface of the cylinder.

Explanation

To find the moment of inertia of a solid cylinder of mass \( M \) and radius \( R \) about a line parallel to the axis of the cylinder and on the surface of the cylinder, we can use the parallel axis theorem. Here’s a step-by-step solution:

### Step 1: Moment of Inertia about the Central Axis
First, we need to find the moment of inertia of the solid cylinder about its central axis (the axis that goes through the center and is perpendicular to the circular face). The formula for the moment of inertia \( I \) of a solid cylinder about its central axis is given by:

\[ I_{\text{central}} = \frac{1}{2} M R^2 \]
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