The Fundamental Theorem of Calculus states:
For a continuous function
with antiderivative
Ptoof: Consider a function
which has antiderivative
and an area function
where
is the area from
to
is clearly an increasing function and
Now consider a narrow strip of the region between
and
The area of this strip is
Since the narrow strip is contained within two rectangles then
hence
and taking the limit as
gives
so the area function is an antiderivtive of
and
differ by a constant at most.
We can write
but
and since
so