The Fundamental Theorem of Integral Calculus

The definite integral ∫_a^bf[x] x is approximated in real terms by taking sums of slices of the form

f[a] · Δx + f[a + Δx] · Δx + f[a + 2Δx] · Δx + ⋯ + f[b^′] · Δx, where b^′ = a + h · Δx and a + (h + 1) · Δx>b

Given a real function f[x] defined on [a, b] we can define a new real function S[a, b, Δx] by

S[a, b, Δx] = f[a] · Δx + f[a + Δx] · Δx + f[a + 2Δx] · Δx + ⋯ + f[b^′] · Δx,

where b^′ = a + h · Δx and a + (h + 1) · Δx>b.  This function has the properties of summation such as

| S[a, b, Δx] | ≤ | f[a] | · Δx + | f[a + Δx] | · Δx + | f[a + 2Δx] | · Δx + ⋯ + | f[b^′] | · Δx

| S[a, b, Δx] | ≤Max[| f[x] | : x = a, a + Δx, a + 2Δx, ⋯, b^′] · (b - a),

We can say we have a sum of infinitesimal slices when we apply this function to an infinitesimal δx,

∫_a^bf[x] x≈Underoverscript[∑, Underscript[x = a, step δx], arg3] f[x] · δx or ∫_a^bf[x] x = st[Underoverscript[∑, Underscript[x = a, step δx], arg3] f[x] · δx], when δx≈0

Officially, we code the various summations with the functions like S[a, b, δx] (in order to remove the function f[x] as a variable.)  We need to show that this is well-defined, that is, gives the same real standard part for every infinitesimal,

S[a, b, δx] ≈S[a, b, ι] and both are limited (so they have a common standard part.)

When f[x] is continuous,  we  can show this  "existence," but in the case of the Fundamental Theorem, if we know a real function F[x] with dF[x] = f[x] dx for all a≤x≤b, 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 F[b] - F[a].  The only ingredient needed to make this work is that

Max[| ε[x, δx] | : x = a, a + δx, a + 2δx, ⋯, b^′] = ε[a + k δx, δx] ≈0

This follows from the Uniform Differentiability Theorem above when we take one of the equivalent conditions as the definition of "dF[x] = f[x] dx for all a≤x≤b."

Notice that ε[x, Δx] is the real function (f[x + Δx] - f[x])/Δx - f^′[x], so we can define an infinite sum by extending the real function

S_ε[a, b, Δx] = Underoverscript[∑, Underscript[x = a, step δx], arg3] | ε | · δx

For more details see p.51-53.

Created by Mathematica  (September 22, 2004)