Example: Uniform rod of length L rotating about one end

Goal: evaluate the moment of inertia integral   I = r2 dm

For a uniform rod of length L rotating about an axis passing perpendicularly through one end of the rod, align the rod along the x-axis. Split the rod into pieces of size dx.

The mass of each piece is:   dm = λ dx, where λ = M/L is the mass per unit length of the rod.

The integral becomes:   I = 0L x2 λ dx = λ x3/3 |0L = λ L3/3

Substituting M = λL gives:

I =
ML2     for a uniform rod rotating about one end.