## Monday, June 10, 2013

### Review of "The Calculus Integral"

Our textbook THE CALCULUS INTEGRAL has just been reviewed in the most recent issue of the American Math Monthly:

The review, by David Bressoud,  covers
• Calculus by Michael Spivak;
• Calculus Deconstructed: A Second Course in First-Year Calculus by Zbigniew Nitecki;
• Approximately Calculus by Shahriar Sharhriari;
• A Guide to Cauchy's Calculus: A Translation and Analysis of Calcul Infinitesimal by Dennis M. Cates;
and
• The Calculus Integral by Brian S. Thomson
I cannot provide a link to the entire review (which should be of interest to anyone following the teaching of integration theory).  But maybe fair use allows me to quote two of the paragraphs of interest to readers of our text:
...``The final book is the most challenging of all, Brian Thomson’s The Calculus Integral. The author introduces it as appropriate for an honors course in calculus, but it would require some very special students. I could see it used for a senior seminar. Every proof, every example, and every counterexample is left as an exercise. The proofs
are scaffolded and answers are provided in the second half of the book, but if approached as intended, working through this text would be daunting. Like Nitecki and Spivak, Thomson focuses on the theorems of calculus. He begins with sequences, the careful derivation of properties of continuous functions, and the definition and basic
properties of differentiation. He takes an abrupt turn when he gets to integration. Like the other four authors, Thomson rejects Riemann’s definition of the definite integral, but he takes it a step further. He returns to Newton’s definition of the integral as antiderivative, what he calls “the calculus integral.” Thomson defines the definite integral
of f as the change in a uniformly continuous function that is an antiderivative of f at all but at most finitely many points. A function is integrable if and only if such an antiderivative exists. The fundamental theorem of integral calculus is now the observation, via the mean value theorem, that for any integrable function f and any partition
of the interval over which we integrate, for each i there exists a value xi in the ith subinterval such that the definite integral is exactly the sum over i of f (xi ) times the length of the ith subinterval. From here, he explores Riemann sums as approximations to definite integrals. Thomson’s handling of integration has the advantage that it clarifies the point of the fundamental theorem of integral calculus, which is the general equivalence of two very different approaches to integration.

There is much more to Thomson’s book. He continues on to the monotone convergence theorem, then to sets of measure zero, to absolute continuity, and on to define the Lebesgue integral of f as the change in a function F that is an antiderivative of f almost everywhere and is absolutely continuous in the Vitali sense. He concludes with
the Henstock–Kurzweil integral. His book is magnificent, but even if we stop before preparing for the monotone convergence theorem, it would be supremely demanding for an honors course in the first year.''

Students and teachers of integration theory should be familiar with David Bressoud  whose two "radical" books are must reads for anyone with a serious interest in this subject.