The Fundamental Theorem of Integral Calculus

Now we use the intuitive microcsope approximation (1.1.1) to prove:

Theorem: The Fundamental Theorem of Integral Calculus: Part 1

Suppose we want to find .  If we can find another function    so that the differential  gives    for every    , then

The definition of the integral we use is the real number approximated by a sum of small slices,

     ,when

Telescoping sums & derivatives

We know that if    has derivative  , the differential approximation above says,

so we can sum both sides

The telescoping sum satisfies,

so we obtain the approximation,

This gives,

or  .  Since    is continuous,  , so  .

We need to know that all the epsilons above are small when the step size is small, ,  when    for all  .  This is a uniform condition that has a simple appearance in Robinson's theory.  There is something to explain here because the theorem stated above is false if we take the usual pointwise notion of derivative and the Reimann integral. (There are pointwise differentiable functions whose derivative is not Riemann integrable.)

The condition needed to make this proof complete is natural geometrically and plays a role in the intuitive proof of the inverse  function theorem in the next section.

Created by Mathematica  (September 22, 2004)