Postscript: A Rigorous Treatment of the Definite Integral

In the year following that on which these notes are based, there was enough time at the end of the course to permit a rigorous discussion of the definite integral. It was pointed out that the concept of area used in our previous definition was only intuitive, and the problem of defining the concept of area under a more or less smooth curve was discussed. This led naturally to approximations above and below by rectangles, or to upper and lower sums. As in Begle, Introductory Calculus, (Holt, New York, 1954), the upper and lower integrals were defined as greatest lower bound and least upper bound, respectively, of the upper and lower sums, and the integral was defined as the common value of the upper and lower integrals when they are equal. The theorems about integrals previously used and the existence theorem for the integral of a continuous function were then proved as in §§3 and 4 of Chapter 7 of the book by Begle cited above.