The Fundamental Theorem of Calculus (FTC) is a theorem par excellence and no calculus student should go throug a first course in Calculus without gaining some insight into nature of this theorem and its applications. In the first few sectins of this chapter, we have studied two basic concepts: antidifferentiation and definite integrals. The fist of these concepts refers tothe process which is exactly the reverse of taking derivatives. The second concept is associated with the process of finding an area under a graph by dividing the area into small rectangles and then taking a limit of a Rieman sum whose evaluation in general entails a formidable task It is most remarkable that these two seemingly unrelated concepts are in fact different aspects of the same idea is the basic content of the FTC.
We make no attempt to present in this module a perfectly rigorous proof of this beautiful theorem, but we hope that the approach that we take, will give the student a fairly good grasp of the central idea which makes this theorem work. There are actually two versions of the FTC and the order in which they are proved is strictly a matter of taste. Each version of the theorem can be easily deduced once the other has been proved.
In this module, what we call version 2 theorem is in fact the version that students will find more useful. We choose to prove the other version first because our geometric approach makes the theorem more intuitive. Throughout the proofs, we will always assume that f(x) is a continuous function over a closed interval and that f(x ) is an antiderivative of a function F(x) over that inverval.
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Gabriel G. Lugo, lugo@uncwil.edu
Russell L. Herman, hermanr@uncwil.edu
Last updated November 29, 1998