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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!

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    This is a definition of "integral." If you mean the idea of using an anti-derivative, now that's another story altogether. – 2017-02-12
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    The latter part of your comment is precisely my question. I guess the answer is beyond my scope rite now. – 2017-02-12
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    @darylnak This is the fundamental theorem of calculus and it's credited to Newton and Leibniz in the 1660s and 70s. If your question is about how they discovered it, then that's a long, fascinating story about which much has been written but I'm not super familiar. If your question is how to understand why the antiderivative and the integral are related in such a way, then that's something that has a quick answer. – 2017-02-12
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    @darylnak Oddly, the definition of an integral through a sum is often called the 'Riemann integral' after Riemann's thesis in the 1850s and clarifications by Darboux in the 1870s, even though the fact that the area under a curve can be thought of as a limiting sum was well known to Newton, and even well before that. The history behind why this is the case is also inevitably very interesting. – 2017-02-12
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    @spaceisdarkgreen Pre C19, there was only one notion of integral, which was the thing that did the opposite of differentiation, and happened to find areas. Cauchy formalised it for continuous functions, using the area notion, in his restructuring of the *Cours d'Analyse* in the 1820s, and Dirichlet, and then Riemann, extended the definition to more functions. It was only later, after non-integrable functions were discovered, that it was deemed inadequate. [...] – 2017-02-19
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    [...] Then Lebesgue proposes an alternative definition, also founded on area, but from a different point of view that turned out to work much better on more functions at the expense of departing from the traditional approach. It then so happened that Riemann's name got attached to it over Darboux, but the real point is that before Lebesgue, there was no reason to attach a further distinction: if you've only got one integral-like thing continuously developed from Newton/Leibniz, there's no reason to label it as anything other than *the* integral. – 2017-02-19

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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.