Yes there is, but it's a pain to do, which is why we don't consider it compared to first principles for differentiation.
I would write it out but it's a bit of a pain with Latex, so see 4:40 of this video:
Yes that's the definition of an integral. An integral is essentially just an infinite sum, so the upper limit of the sum needs to go to infinity, otherwise it just becomes an approximation. An alternatively way to write it is:
where its understood that you increment the summation in units of rather than units of 1.