The Fundamental Theorem of Integral Calculus
The definite integral is approximated in real terms by taking sums of slices of the form
, where and
Given a real function defined on we can define a new real function by
,
where and . This function has the properties of summation such as
,
We can say we have a sum of infinitesimal slices when we apply this function to an infinitesimal x,
or , when
Officially, we code the various summations with the functions like (in order to remove the function as a variable.) We need to show that this is well-defined, that is, gives the same real standard part for every infinitesimal,
and both are limited (so they have a common standard part.)
When is continuous, we can show this "existence," but in the case of the Fundamental Theorem, if we know a real function with for all , the proof in Section 1 interpreted with the extended summation functions and extended maximum functions proves this "existence" at the same time it shows that the value is . The only ingredient needed to make this work is that
This follows from the Uniform Differentiability Theorem above when we take one of the equivalent conditions as the definition of " for all ."
Notice that is the real function , so we can define an infinite sum by extending the real function
For more details see p.51-53.
Created by Mathematica (September 22, 2004)