Monday, 21 January 2013

calculus - Definite Integral of a infinitesimal



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 xϵ:[a,b]R. The value of the expression is (ba)ϵ.


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