## The Fundamental Theorem of Calculus

Let be a continuous function on the interval . The following theorem is incredibly useful in mathematics, physics, biology, etc.

Theorem 2.1.3   If is any differentiable function on such that , then One reason this is amazing, is because it says that the area under the entire curve is completely determined by the values of a (magic'') auxiliary function at only points. It's hard to believe. It reduces computing (2.1.2) to finding a single function , which one can often do algebraically, in practice.Whether or not one should use this theorem to evaluate an integral depends a lot on the application at hand, of course. One can also use a partial limit via a computer for certain applications (numerical integration).

Example 2.1.4   I've always wondered exactly what the area is under a hump'' of the graph of . Let's figure it out, using . But does such an always exist? The surprising answer is yes''.

Theorem 2.1.5   Let . Then for all .

Note that a nice formula'' for can be hard to find or even provably non-existent.

The proof of Theorem 2.1.5 is somewhat complicated but is given in complete detail in Stewart's book, and you should definitely read and understand it.

Proof. [Sketch of Proof] We use the definition of derivative.    Intuitively, for sufficiently small is essentially constant, so (this can be made precise using the extreme value theorem). Thus which proves the theorem. William Stein 2006-03-15