What is DRIP?          DUMP-THE-RIEMANN-INTEGRAL-PROJECT (D.R.I.P.)   Top ten reasons for dumping the Riemann integral:  #10, #9, #8, #7, #6, #5, #4, #3, #2, #1 A number of academic mathematicians feel that the Riemann integral no longer belongs in an undergraduate course of instruction.    Here is a brief if quirky 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; the former is focussed largely on Lebesgue's theory of measure and integration and the latter includes accounts only of the Riemann integral and the improper integral.   An experimental dripped version, however, of our elementary text is available; you can now  download a copy of this or purchase a trade paperback.  THE CALCULUS INTEGRAL and the THEORY OF THE INTEGRAL on this website do present theories of the Newton and Henstock-Kurzeil integrals in addition to the Riemann theory.   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 Dieudonne,   J. Dieudonné, Foundations of Modern Analysis. (Pure and Applied Mathematics, Vol. X) XIV + 316 S. New York 1960. Academic Press Inc.   "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."   The solution to this problem that this author proposed was the teaching of the 'regulated integral.'  There is a brief outline of that integral in our textbook  THEORY OF THE INTEGRAL on this website. Another account of the regulated integral and a goodpedagical discussion was given by Sterling Berberian in this Monthly Article.   Jean Dieudonne (1906-1992) was one of the great mathematicians of the twentieth century.  The account by Pierre Cartier sums it up well. In the 1950s two mathematicians independently introduced a simple modification of the Riemann integral that completely described the standard integration theory on the real line, including the Riemann integral, the improper Riemann integral, the Lebesgue integral, and the Denjoy-Perron integral.  It is now considered by many mathematicians that the integral described by Ralph Henstock and Jaroslav Kurzweil could be the basis for a better instruction for undergraduate students of all the traditional elements of integration theory.    This should by no means be interpreted as abandoning any of the elements of the Lebesgue theory of integration:  it is simply a proposal to introduce those same elements in some more intuitive order and, at the same time, include a full account of the nonabsolute integration theory.   Kurzweil (1926-    ) is still alive at this time of this post, although well into his eighties.  A report on his career with many kind words about him personally was published by his Czech colleages in 2006.   Henstock (1923-2007) died a short while ago.  His student Pat Muldowney published a nice Obituary Notice for him in the Bulletin of the London Math. Society. ... from  Robert G. 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)."   See also the article Robert G. Bartle,  Return to the Riemann Integral Amer. Math. Monthly 103 (1996), no. 8, 625–632.   A biographical essay on the late Bob Bartle (1927-2003)  is also worth viewing.       SOME FURTHER DRIP RESOURCES   Several or our textbooks  include an account of the Henstock-Kurzweil integral (all of the texts Elementary Real Analysis: Dripped Version, The Calculus Integral, Theory of the Integral, and Real Analysis include material on this integral, usually along with other ideas in integration theory).  In addition the reader is referred to a number of other sources that can be used.     Elementary Real Analysis: The Dripped Version  Authored by Brian S. Thomson, Judith B. Bruckner, and Andrew M. Bruckner  List Price:$19.95 7" x 10" (17.78 x 25.4 cm) Black & White on Cream paper 596 pages ISBN-13: 978-1438248509(CreateSpace-Assigned) ISBN-10: 1438248504 BISAC: Mathematics / Mathematical Analysis Preliminary price of $15.00 if you ask the author for a DISCOUNT CODE. This is a "Work In Progress" and later printings will have a number of additions and corrections. A relatively current and free PDF file is available for free download.    A first course in elementary real analysis, with the modern integration theory introduced rather than the unfortunate Riemann integration.[D.R.I.P.=Dump the Riemann Integral]   CreateSpace eStore: https://www.createspace.com/3346154 ... more info.        Download free hyperlinked PDF file.         The Calculus Integral   Authored by Brian S Thomson  List Price:$14.95 7" x 10" (17.78 x 25.4 cm) Black & White on Cream paper 304 pages ISBN-13: 978-1442180956(CreateSpace-Assigned) ISBN-10: 1442180951 BISAC: Mathematics / Calculus An elementary introduction to integration theory on the real line. This is at the level of an honor's course in calculus or a first undergraduate level real analysis course. In the end the student should be adequately prepared for a graduate level course in Lebesgue integration.           CreateSpace eStore: https://www.createspace.com/3384432 ... more info.      Download free hyperlinked PDF file.         Theory of the Integral    Authored by Brian S Thomson 6" x 9" (15.24 x 22.86 cm) Black & White on Cream paper 390 pages ISBN-13: 978-1467998161(CreateSpace-Assigned) ISBN-10: 1467998168 This text is intended as a treatise for a rigorous course introducing the elements of integration theory on the real line. All of the important features of the Riemann  integral, the Lebesgue integral, and the Henstock-Kurzweil integral are covered. The text can be considered a sequel to the four chapters of the more elementary text THE CALCULUS INTEGRAL which can be downloaded from our web site. For advanced readers, however, the text is self-contained.   Prepublication: planned for June 2012 ... more info.   Download free hyperlinked PDF file. This file is being updated frequently until publication date.