DUMP-THE-RIEMANN-INTEGRAL-PROJECT (D.R.I.P.)
Here is a brief
HISTORY of the project. If you think that your students (and you) have not been confused by the standard undergraduate and graduate education in integration theory please take this
QUIZ. We provide this information here to keep the real analysis community up-to-date on the project. Our two texts REAL ANALYSIS and ELEMENTARY REAL ANALYSIS follow the normal, mainstream trend and include accounts of the Riemann integral and the improper integral. An experimental dripped version of our elementary text is in preparation; you can now download a copy of this, but do consider it a work in progress.
[Alan Smithee has also offered us his notes for this purpose and they are available on this web site. See the MENU item.]
Please send us any information about whether this project has had an impact on your teaching or the teaching of any of your colleagues.
 | ...from Jean Dieudonn\'{e}, Foundations of Modern Analysis, (1960) Finally, the reader will probably observe the conspicuous absence of the 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. 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, 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. When one needs a more powerful tool there is no point in stopping halfway, and the general theory of (Lebesgue) integration is the only sensible answer. |
 | ... from Robert Bartle's review of the monograph The general theory of integration, by Ralph Henstock. Oxford Mathematical Monographs, Clarendon Press, Oxford, 1991, xi 262pp., ISBN 0-19-853566-X In elementary calculus courses we are usually successful in teaching students to evaluate an integral of a suitable function f = F' on an interval [a, b], by evaluating F(b)−F(a), but we are often not very successful in connecting this type of integration with Riemann sums and their limits. During their junior/senior year, students who are studying mathematics seriously are then led through a more careful and exhaustive discussion of these ideas. However, they are informed that all of this is only tentative, since when they become graduate students they will replace the outmoded Riemann integral that they have just mastered with the Lebesgue integral. Of course, it is not completely replaced by this new integral, because there are certain notions, such as "improper integrals", that do not fall under this new umbrella and are still of considerable importance; moreover, almost all evaluations of integrals (whether Riemann or Lebesgue) are found by using the F(b) − F(a) method, with a few minor variations. We tell our advanced undergraduates that we would like to introduce them to the Lebesgue integral but cannot do so since it requires a prior study of measure theory and/or topology and is "too advanced" for them at their present stage of mathematical study. Probably none of us is satisfied by this circuitous procedure. Suppose that someone came up with an approach to the integral that simultaneously covered the integration of all functions that have antiderivatives, all functions that have Riemann integrals, all functions that have improper integrals, and all functions that have Lebesgue integrals. Moreover, suppose that the definition of this "superintegral" was only slightly more complicated than that of the Riemann integral, that its development required no study of measure theory, no study of topology, and that this integral had properties that correspond to the Monotone Convergence Theorem and the Lebesgue Dominated Convergence Theorem (among others). If this mathematical miracle occurred, then wouldn’t this new approach be immediately adopted, at least at the junior/senior level course, and quickly worked into the calculus level? The answer is a resounding: No! Proof. In fact, such an integral has already been developed and has been around for some time, but its existence has remained largely unknown (except to readers of the Real Analysis Exchange) and it has had very little, if any, educational impact (known to this reviewer). |
