> 'Horizontal integrals' does not work. It refers to one way of visualising functions (and even then, only of 1 or 2 variables, and only with specific conventions), and not to the integral itself.
We already have "left" and "right" Riemann sums...
Well, yes, it’s bad, I am not defending it (and I have never seen it in the wild, even in class when I was taught Riemann integration). I am not sure how the supposed popularity of ‘right Riemann sum’ means that ‘horizontal integral’ would be any good.
I guess I'm saying it's not bad; you're just declaring it as such. People find them quite intuitive. I don't know about you but I've literally never seen someone get confused over silly things like "why is it called a left sum if I can plot the function in the other direction".
OTOH I’ve literally never seen someone using that phrase but obviously YMMV. It’s useful as a teaching aid and that’s about it. It’s easy to come up with examples of bad nomenclature used in the wild so I’m not sure what your point is. If you think it’s fine, then, more power to you. It wasn’t about that.
We already have "left" and "right" Riemann sums...