As the following math stack exchange question (and answers) show: https://math.stackexchange.com/questions/703212/is-dxdy-really-a-multiplication-of-dx-and-dy There are a lot of different ways to think about the integral.

  • Any ideas about how the concept of the integral should be taught, with special reference to teach an understanding of the relationship between the different conceptual ways to define "integral" (thus, we are not talking about a first course teaching integration here)?

  • Also, good references for comparison of different concepts of the integral would be nice.

Edit: As an answer to the question in the comment: The question linked to above is about the meaning of the symbol "$dx$" in the integral. I have in mind various notions related to the integral such as differential forms, Riemann sums, measures, ...

What are relationship between Integral defined via differential forms, Riemann sums, measures, as a linear operator (Daniel integral?), geometric integral, and so on. Especially relationship between this various forms of integral, and comparisons between them.


2 Answers 2


At least as a counter-point to other reasonable viewpoints: I am not such an enthusiast about measure theory as "foundation" for integrals, because it seems to me that measure theory over-emphasizes just one smallish part of what we really want. Namely, starting with (ambiguously) nice functions $f_n$, perhaps continuous and compactly supported on a Euclidean space, so that any notion of integral makes sense for the individuals, _one_thing_ we want is that $\lim_n \int f_n=\int\lim_n f_n$. Part of the issue is in what sense the limit of the functions themselves is taken, and there is not a unique answer to this. But it is clear from examples that pointwise limits of continuous functions may be harder to sensibly integrate in naive terms.

Another fundamental issue is the distinction between integrating differential forms and integrating functions, although again I claim this is more of a technical distinction than fundamental. Namely, while from a "measure-theory" viewpoint integrating $f$ on an interval $[a,b]$ does not even mention direction, in usual calculus, and if we imagine we are integrating the differential form $f\,dx$, direction does matter. (That is, in elementary calculus the integral $\int_a^b$ is the negative of $\int_b^a$.) This issue is still more notational than substantive even when doing path integrals and surface integrals. It seems to me that the issue is serious only when we are doing calculus on manifolds, where there is no canonical coordinate system, so that a differential form has no canonical local expression, and the only thing that is well-defined is an integral of an exterior $k$-form over a $k$-cycle or $k$-current, etc.

Still, in all cases, the external properties and desiderata are very similar, so that all these things are more the same than different.

Even when (as in basic Lebesgue theory) we discover that some restrictions are necessary to be able to pass a linear operator inside an integral, it seems to me that the development of mathematics has been to make sense of otherwise-dubious outcomes, whenever possible. For example, $\int_{\mathbb R} {e^{ix\xi}\;dx\over 1+x^2}$ cannot be differentiated with respect to $\xi$ very much without producing a not-classically-sensible integral. But as Fourier transform of a tempered distribution it's still completely fine.


For some references on the differential forms approach, the following two math.SE questions might be helpful (because of their references): https://math.stackexchange.com/questions/714646/construction-of-the-hyperreal-numbers, https://math.stackexchange.com/questions/315768/how-to-create-the-set-of-hyperreal-numbers-using-ultraproduct.

In general, if you want to understand the integral, I would suggest starting with measure theory. Understanding abstract measure theory will be like turning on a light that illuminates all of integration theory. In this setup, the meaning of the $dx$ becomes clear. So, in my opinion one should start with measure theory.

You might look at this question: Best textbooks to introduce measure theory and Lebesgue integration? for references.

  • $\begingroup$ I indeed agree that one should start with measure theory, and I have done that. I have a reasonably good knowledge of measure theory, what I do want know is a treatment that compares the different approaches. For an example of what I am after, see en.wikipedia.org/wiki/Improper_integral , subsection Types of integrals. $\endgroup$ Commented Mar 17, 2014 at 11:35
  • 1
    $\begingroup$ I'm confused by the first sentence of this answer, because it talks about differential forms, but the links deal with NSA. $\endgroup$
    – user507
    Commented May 19, 2014 at 22:22

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