1. The problem statement, all variables and given/known data This is supposed to be a proof of the fundamental theorem of calculus. I'm not really sure what that proves, but to me at least it does not prove that the area under a curve is the antiderivative of the function and then inserting the upper x value and subtracting from it the lower x value. I'm sort of convinced that to find the area beneath a function you take the antiderivative. For instance if your velocity is constant at 50 mph and you travel 2 hours, then you take the antiderivative of that, 50x, then plug in 2 for x, you get 100 miles and subtract that from zero and you still get 100. Are they just proving by analogy that since the antiderivative works for simple geometric shapes for which the area is known that it must work for other more complex functions? I also understand how the infinitesimal sections of an area cancel in the middle. If you have an interval from a to z, and you keep adding in subintervals, say, (a-b) + (c-b) + (d - c) + (z - d) I understand how those cancel except z - a. However, I'm still not fully convinced that to find the area beneath a curve you take the antiderivative of the function and subtract the upper x interval from the lower x interval. I don't see how the proof above shows that.