Proof of the Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus states:

For a continuous functionwith antiderivative

Ptoof: Consider a functionwhich has antiderivativeand an area functionwhereis the area fromto

is clearly an increasing function and

Now consider a narrow strip of the region betweenand

The area of this strip isSince the narrow strip is contained within two rectangles then

henceand taking the limit asgivesso the area function is an antiderivtive ofanddiffer by a constant at most.

We can write

butand sinceso