In calculus courses we learn what integrals are and how to use them to compute areas, volumes, work and other quantities which are useful and interesting. The calculus sequence, and frequently the whole of the undergraduate mathematics program, does not reach the most powerful theorems of integration theory. I believe that the generalized Riemann integral can be used to bring the full power of the integral within the reach of many who, up to now, get no glimpse of such results as monotone and dominated convergence theorems. As its name hints, the generalized Riemann integral is defined in terms of Riemann sums. It reaches a higher level of generality because a more general limit process is applied to the Riemann sums than the one familiar from calculus. This limit process is, all the same, a natural one which can be introduced through the problem of approximating the area under a function graph by sums of areas of rectangles. The path from the definition to theorems exhibiting the full power of the integral is direct and short.

I address myself in this book to persons who already have an acquaintance with integrals which they wish to extend and to the teachers of generations of students to come. To the first of these groups, I express the hope that the organization of the work will make it possible for you to extract the principal results without struggling through technical details which you find formidable or extraneous to your purposes.