Deriving the moment of inertia of a thin rod.

1. The problem statement, all variables and given/known data

I need to derive the moment of inertia of a thin rod with its axis of rotation at the end of the rod.
The third one.

2. Relevant equations
I = mr^2

3. The attempt at a solution
I completely understand how the moment of inertia is mL^2/3, but what I don’t understand is why linear density is used. M/L. Shouldn’t it be volume density, or area density at the very least since a thin rod is 3D?

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