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Unless otherwise stated, assume the density is constant

**Problem Statement:**

Calculate the rotational inertia of a long, thin rod of length \( L \) and mass \( M \) about one end. Assume the density of the rod is given by

\[
\lambda = \lambda_0 \left(1 + \frac{x}{L}\right)
\]

What is the rotational inertia (about one end) of this rod?

**Explanation of the Formula:**

- \( \lambda \) represents the linear density of the rod, which varies along its length.
- \( \lambda_0 \) is the initial linear density.
- \( x \) is the position along the rod.
- \( L \) is the total length of the rod.

**Question:**

What is the rotational inertia of this rod when rotated about one of its ends?

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