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 ϵ be an infinitesimal as described by Abraham Robinson. Consider the expression:
∫baϵ
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
It makes as much sense as, say, ∫ba2 does — or ∫ba2dx if the former looks too weird. As with the example just shown, in ∫baϵ you're using ϵ as a shorthand for the constant function x↦ϵ:[a,b]→∗R. The value of the expression is (b−a)ϵ.
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