Preface
There are plenty of calculus books available, many free or at least cheap, that discuss integrals. Why add another one?
Our purpose is to present integration theory at an honors calculus level and in an easier manner by defining the definite integral in a very traditional way, but a way that avoids the equally traditional Riemann sums definition.
Riemann sums enter the picture, to be sure, but the integral is defined in the way that Newton himself would surely endorse. Thus the fundamental theorem of the calculus starts off as the definition and the relation with Riemann sums becomes a theorem (not the definition of the definite integral as has, most unfortunately, been the case for many years).
As usual in mathematical presentations we all end up in the same place. It is just that we have taken a different route to get there. It is only a pedagogical issue of which route offers the clearest perspective. The common route of starting with the definition of the Riemann integral, providing the then necessary detour into improper integrals, and ultimately heading towards the Lebesgue integral is arguably not the best path although it has at least the merit of historical fidelity.
Acknowledgments
I have used without comment material that has appeared in the textbook
[TBB] Elementary Real Analysis, 2nd Edition,
B. S. Thomson, J. B. Bruckner, A. M. Bruckner,
ClassicalRealAnalyis.com (2008).
I wish to express my thanks to my co-authors for permission to recycle that material into the idiosyncratic form that appears here and their encouragement (or at least lack of discouragement) in this project.
I would also like to thank the following individuals who have offered feedback on the material, or who have supplied interesting exercises or solutions to our exercises: [your name here], . . .
