The theory and application of integrals is one of the great and central themes of mathematics. For applications, one often requires numerical values. The prob-
lem of numerical integration is therefore a basic problem of numerical analysis.
It is often an elementary component in a much more complex computation.
Numerical integration is, paradoxically, both simple and exceedingly difficult.
It is simple in that it can often be successfully resolved by the simplest of
methods. It is difficult in two respects: first, in that it may require an inordinate
amount of computing time, verging in some unfavorable situations toward im-
possibility; second, in that one can be led through it to some of the deepest
portions of pure and applied analysis. And not only to analysis but, in fact, to
diverse areas of mathematics; for inasmuch as the integral has successfully
invaded many areas, it is inevitable that from time to time these areas turn about
and contribute their special methods and insights to the problem of how integrals
may be computed expeditiously. The problem of numerical integration is also
open-ended; no finite collection of techniques is likely to cover all the possibili-
ties that arise and to which an extra bit of ingenuity or of special knowledge may
be of great assistance.