I know integrals are defined as the following:
$$\lim_{n\to\infty} \sum\limits_{k=1}^{n} f(c_k) \Delta x = \int_{a}^{b} f(x) dx$$
My question is how did someone figure out that the anti-derivative of a function gives the area under the curve? The limit of the sum makes perfect sense (I'm only a calc 2 student).
Sorry if it's suppose to be intuitive!
It was well known by self intution that in order to find the value of an unknown volume, you could just make a sum of known values that could fit into the wanted one. Using this, circle area and parabola area's approximations were discovered.
Integrals as a whole new chapter in math came when the fundamenthal theorem of calculus was proposed by Newton and Leibniz, but again, this came from infinite assumptions which were later proved true.