When I did further maths at college, we spent a couple of hours on a particular kind of integration, where the function was integrated with respect to the length of the path along the function, typically starting at the point x = 0, y = f(0), and typically calling the path length variable s. I remember that this was curious for all sorts of reasons, but not any specific reason.
It may have been called implicit integration, but googling for that phrase seems to suggest I am remembering it wrong.
I get puzzled looks every time I describe this to people who have studied maths for years for some reason. Does this sound familiar and if so, what's the common name for this?