The Calculus Integral [Beta0.2],
B S Thomson,
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Well "calculus book" doesn't quite describe it. It is an account of integration theory on the real line that starts in the initial chapters with the "calculus integral" (i.e., the original integral of Newton) and carries the development as far as the integrals of Lebesgue and Henstock-Kurzweil.
If you are, however, teaching a calculus course and have ever considered dropping the Riemann integral and its ugly step-sister the improper Riemann integral from the syllabus, then you might want to use the first three chapters as a basis for the teaching of the basic integral of the calculus.
A trade paperback version will be available shortly.