Fundemental theorem of Calculus intuition

[Jehanne]

I've always been curious why the fundamental theorem of calculus works, I'm talking about the one which talks about the derivative of an integral of a function gives the function itself. Why does this happen? Am I missing something completely obvious?

In other words, why does the slope of the tangent line of every point on an area function result in the original function?

While not entirely related to your question, here's a great video on the underlying theory:

http://www.dailymotion.com/video/x2vyuf8