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
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 $(b-a)\epsilon$.
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