In the introduction to chapter VIII of Dieudonné's *Foundations of Modern Analysis* (Volume 1 of his 13-volume *Treatise on Analysis*), he makes the following argument:

Finally, the reader will probably observe the conspicuous absence of a time-honored topic in calculus courses, the “Riemann integral”. It may well be suspected that, had it not been for its prestigious name, this would have been dropped long ago, for (with due reverence to Riemann’s genius) it is certainly quite clear to any working mathematician that nowadays such a “theory” has at best the importance of a mildly interesting exercise in the general theory of measure and integration (see Section 13.9, Problem 7). Only the stubborn conservatism of academic tradition could freeze it into a regular part of the curriculum, long after it had outlived its historical importance. Of course, it is perfectly feasible to limit the integration process to a category of functions which is large enough for all purposes of elementary analysis (at the level of this first volume), but close enough to the continuous functions to dispense with any consideration drawn from measure theory; this is what we have done by defining only the integral of regulated functions (sometimes called the “Cauchy integral”). When one needs a more powerful tool, there is no point in stopping halfway, and the general theory of (“Lebesgue”) integration (Chapter XIII) is the only sensible answer.

I've always doubted the value of the theory of Riemann integration in this day and age. The so-called Cauchy integral is, as Dieudonné suggests, substantially easier to define (and prove the standard theorems about), and can also integrate essentially every function that we might want in a first semester analysis/honors calculus course.

For any other sort of application of integration theory, it becomes more and more worthwhile to develop the fully theory of measure and integration (this is exactly what we did in my second (roughly) course on analysis, so wasn't the time spent on the Riemann integral wasted?).

Why bother dealing with the Riemann (or Darboux or any other variation) integral in the face of Dieudonné's argument?

Edit: The Cauchy integral is defined as follows:

Let $f$ be a mapping of an interval $I \subset \mathbf{R}$ into a Banach space $F$. We say that a continuous mapping $g$ of $I$ into $F$ is a primitive of $f$ in $I$ if there exists a denumerable set $D \subset I$ such that, for any $\xi \in I - D$, $g$ is differentiable at $\xi$ and $g'(\xi) =f(\xi)$ .

If $g$ is any primitive of a regulated function $f$, the difference $g(\beta) - g(\alpha)$, for any two points of $I$, is independent of the particular primitive $g$ which is considered, owing to (8.7.1); it is written $\int_\alpha^\beta f(x) dx$, and called the integral of $f$ between $\alpha$ and $\beta$. (A map $f$ is called regulated provided that there exist one-sided limits at every point of $I$).

Edit 2: I thought this was clear, but I meant this in the context of a course where the theory behind the integral is actually discussed. I do not think that an engineer actually has to understand the formal theory of Riemann integration in his day-to-day use of it, so I feel that the objections below are absolutely beside the point. This question is then, of course, in the context of an "honors calculus" or "calculus for math majors" course.

answerpedagogical questions, I don't think it's fair to jump on somebodyaskinga question on teaching for not having enough experience. $\endgroup$ – Dan Petersen Jan 21 '11 at 10:37