Our textbook

*has just been reviewed in the most recent issue of the American Math Monthly:*

**THE CALCULUS INTEGRAL***Table of Contents*June/July 2013

**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;

*The Calculus Integral*by Brian S. Thomson

*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’sThe 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

offas the change in a uniformly continuous function that is an antiderivative offat 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 functionfand any partition

of the interval over which we integrate, for eachithere exists a valuexiin theith subinterval such that the definite integral is exactly the sum overioff (xi )times the length of theith 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 offas the change in a functionFthat is an antiderivative offalmost 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.

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