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?
I've found it! It was called an intrinsic equation, and presumably we integrated it to obtain a cartesian representation.
Perhaps there was no special term for the integration; maybe it was something like "Integrating an intrinsic curve".
There are obviously a few things I got wrong in the question, which made answering it a bit of a guesswork.