A non-uniform rod AB has a mass `M` ad length 2l. The mass per unit length of the rod is mx at a point of the rod distant x from A. find the moment of inertia of this rod about an axis perpendicular to the rod (a) through A (b) through the mid-point of AB.

Explanation

To find the moment of inertia of a non-uniform rod AB with mass \( M \) and length \( 2l \) about two different axes, we will follow these steps:

### Part (a): Moment of Inertia about an Axis through A

1. **Define the mass per unit length**: The mass per unit length of the rod is given as \( mx \), where \( m \) is a constant and \( x \) is the distance from point A.

2. **Express the mass element**: The mass of a small element \( dx \) at a distance \( x \) from A can be expressed as:
\[
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