(Before I start, I don't know which tag is suitable for this post. Please retag my post if it bothers you.)
Let's say there is a string on $[0,1]$ with a mass given by $m(x)$. ($m(x)$ means the mass contained in the interval $[0,x]$.)
Then the inertia momentum of this string is given by,
$$\int_0^1 x^2 dm(x).$$
(This is a Riemann Stieltjes Integral.)
If $m$ is differentiable on $[0,1]$, this is exactly the same as $\int_0^1 x^2 m'(x) dx$.
I thought the definition of 'density at a point' is given by 'derivative of mass', but I think this definition is not sufficient for many cases.
Is there a precise definition of density at a point?