What is "superficial density" in the context of integration (over surfaces)?

Question

I have read something like this:

We have $$\int_{M}f(x)d\sigma(x) = 1$$ where $d\sigma$ is the superficial density.

What does this mean? $M$ is a hypersurface/manifold. The author does not define it.

Answer 1

Wikipedia article Area density begins with

The area density (also known as areal density, surface density, or superficial density) ...

In mathematics, it is more common to call this "surface measure".