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.

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

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]

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.

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.

Olypian quarrels et gorilla congolium sic ad nauseum.

Olypian quarrels et gorilla congolium sic ad nauseum.

Olypian quarrels et gorilla congolium sic ad nauseum.

Olypian quarrels et gorilla congolium sic ad nauseum.

Olypian quarrels et gorilla congolium sic ad nauseum.

Olypian quarrels et gorilla congolium sic ad nauseum.

Olypian quarrels et gorilla congolium sic ad nauseum.

Olypian quarrels et gorilla congolium sic ad nauseum.

Olypian quarrels et gorilla congolium sic ad nauseum.

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03/05/07

Pro quo hic escorol. Olypian quarrels et gorilla congolium sic ad nauseum.

03/05/07

Pro quo hic escorol. Olypian quarrels et gorilla congolium sic ad nauseum.

03/05/07

Pro quo hic escorol. Olypian quarrels et gorilla congolium sic ad nauseum.