The fundamental theorem of calculus states that for a continuous function on an interval , the integral is both continuous and differentiable on . More specifically, it states that for all in This Demonstration helps to provide the intuition behind this idea.
You may recall that the derivative of a function is defined to be . For arbitrarily small values of , this says , or (A)
We use this idea to help develop the fundamental theorem of calculus. Define to be the area function, that is, . Then the Demonstration shows that and so by (A), (since the approximation is in fact an equality as we take the limit as ). Thus is the derivative of the area function, that is, as desired.