[nimonian]

If you think of it as being an accumulator function it can feel a bit more natural - the _definition_ of this accumulator is that, F(x) is the area from 0 to x

The fact that the derivative of this accumulator function is equal to the original function, this is the fundamental theorem of calculus, and I violently agree with you that this part is shockingly, unexpectedly beautiful

[mchinen]

Thank you for wording it better than I could. On the computational side, the integral as a function that measures how the another function accumulates intuitively lets you measure the area under the curve between two points only requires two variables.

Where it gets interesting is that if I were to naively construct this accumulator in a non-computational, infinitesimal manner, I would probably start using the idea that the infinitesimal accumulation is related to the instantaneous slope and using that somehow with the previous accumulation. But this is going the opposite direction of derivatives that the fundamental theorem uses.

I understand there are a few other types of integrals that slice things different ways that might be more intuitive. I vaguely recall some (non fundamental theorem) proof that made Riemann integrals click for a while.