Why is the anti derivative defined as the area under a curve?

[Millan]

A derivative can be used to find rates of change, slopes, relative extrema etc. (at least from what I know). I don't understand how working backwards and finding the integral of a function suddenly gives the area under a curve. I just don't see the connection and I don't think my teacher does either.

[ricki]

Hello,

This is a question I used to ask myself and teachers all the time till I went to the head of the mathematics department.

This is more of a rule then why it happens. You just have to except it. It is an operation just like x^n/x^v=x^n-v. Honestly, it is one of the beautiful things that happen in math and one of the most interesting as well. When mathematicians started doing derivatives they didn't realize that an anti-derivative was equal to the area under the function as well until Riemann came around and showed Riemann sums and that if you did the opposite operation of a derivative then it is proportional to the area under a certain curve.

The simple answer is that it is something that just so happens in mathematics and that it is just an other operation. It is quiet a beautiful operation as well