http://www.math.wisc.edu/~keisler/calc.html

You define the “area under a curve” function in a similar way to arclength as follows: You show that, given , there is a unique function (to be interpreted as the area under the curve from to such that: for all , and, if is linear on , then is the appropriate value (i.e., the formula for the area of a trapezoid: . You can then prove that there is a unique function with these properties, and it’s .

The way area is dealt with in smooth infinitesimal analysis (i.e., isolate some conditions that an area function must have, then prove there is a unique function satisfying those conditions) is not so different from what is done in the classical case. The problem is that it’s done over again for each problem type: that is, you use it once to determine the area of a cone, then once again to prove the fundamental theorem of calculus, etc.

If I understand your question right, you’re wondering if you can do it once and for all. That is, can you prove that there is a unique function assigning to subsets of, say, an element of satisfying the appropriate conditions. I don’t know, but I would guess that the answer is “no.” I would have to think some more about that (and probably learn some more first!). If you have any thoughts, let me know.

I guess my next question would be: how does one define the “area under a curve” function? I assume we would like to do it without limits.

Now that I look over the above again, I notice that there’s something similar with the arc length function: you mention that it’s not formally defined, so instead work with what reasonable properties it should have. But there has to be some nice way to formally define these notions; perhaps I just need to get Bell’s book.

As a side note, I’m amazed by how much more elegant some of the proofs with smooth infinitesimal analysis are than with regular calculus. In particular, the proofs of the product and chain rules are so much nicer than their usual counterparts.

Axiom 5 doesn’t really axiomatize the fundamental theorem of calculus, since all Axiom 5 says is that antiderivatives of functions exist. The fundamental theorem of calculus says something more: it says that not only do antiderivatives of (suitable) functions exist, but that furthermore the function giving the (signed) area under the graph of from to is a specific example of an antiderivative. (Also, that all other antiderivatives are equal to plus some constant.)

The notation is confusing though, because in Axiom 5, I said that antiderivatives of functions exist, and furthermore I denoted the antiderivative of which takes the value on input by . This could be confusing, because normally is defined as a Riemann sum, which is the classical formalization of the “area” concept, and so with that notation, the statement that the derivative of is is indeed the fundamental theorem of calculus. But with the notation I used in this article, it’s just a definition.

As an aside, you can prove a version of the fundamental theorem of calculus in smooth infinitesimal analysis, in much the same way that you can find the area of a cone.

If you want to not take the existence of antiderivatives as an axiom, you might want to take a look at Bell’s book “A Primer on Smooth Infinitesimal Analysis,” which does a lot of stuff without that axiom, but in its place he assumes that if and then .

However, there is something that bothers me about this setup, and I wonder if you have any ideas on it. Specifically, the problem is Axiom 5. This axiomatizes the fundamental theorem of calculus! Is there no way to define an area function using infinitesimals, then show that its derivative is the original function?

I think the idea of taking as an axiom “every curve is made up of infinitesimally small line segments” a beautiful concept, but it seems to me to be giving up too much to also have to axiomatize the existence of an anti-derivative.

Do you know of any way to not have to take this as an axiom?