Definite Integral of a infinitesimal

Question

I did not study math, but have some foundations in it. I have been looking through some books on nonstandard analysis, and have (what I consider to be) a pretty simple question which I haven't been able to answer through my reading thus far.

Let $\epsilon$ be an infinitesimal as described by Abraham Robinson. Consider the expression:

$\int_{a}^{b} \epsilon$

1) Does this expression even make sense?

2i) If it does make sense, is there a way of calculating what it evaluates to?

2ii) If it doesn't make sense, is there another (rigorous) discipline which can evaluate the quantity?

I would greatly appreciate any direct answers or references to (reasonably easy to read) materials.

Answer 1

It makes as much sense as, say, $\int_a^b 2$ does — or $\int_a^b 2 \rm{d}x$ if the former looks too weird. As with the example just shown, in $\int_a^b \epsilon$ you're using $\epsilon$ as a shorthand for the constant function $x\mapsto\epsilon\colon[a,b]\to {^*}\Bbb R$. The value of the expression is $(b-a)\epsilon$.

Answer 2

This is not quite correct, because integral as it is usually defined is applied to an ordinary real function and produces (in the case of a definite integral) always a real number. So it is not merely a difference of notation between the sloppy notation $\int_a^b 2$ and the more careful notation $\int_a^b 2dx$.

Infinitesimals (and especially infinitesimal partitions) are ordinarily used in defining definite integrals in a fashion that is intuitively appealing and is closer procedurally to what the inventors of the calculus (like Leibniz and Euler) were doing, but they are usually only an intermediate step, and tend to disappear when the final answer is produced. This is true for the definition of the derivative as much as the definition of the integral.

One can further generalize this to apply to more general functions (such as the constant function equal to an infinitesimal $\epsilon$ as you envisioned) but such a generalisation is more delicate and usually involves an understanding of the notion of an internal set. This properly belongs in analysis in Robinson's framework and pretty much does not occur in the calculus.

In short, you typically won't need such integrals when studying the calculus but if you would like to understand Robinson's framework in more detail, the starting point will probably be the notion of an internal set.