Integration

The numerical computation of definite integrals is one of the oldest problems in mathematics. The problem, which in its earliest form involved finding the area of regions bounded by curved lines, has been around for thousands of years, long before the concept of integrals in the framework of analysis was developed in the 17th and 18th century. Certainly the best-known example of this problem was that of computing the area contained in a circle, which in turn led to a study of the number π and its computation. Using a numerical method involving the approximation of a circle by inscribed and circumscribed polygons, Archimedes (287–212 B.C.) was able to give the astonishingly good bounds $$3\frac{{10}}{{71}} < \pi < 3\frac{1}{7}$$. For more on this, see Chap. 5 of the book Numbers by H.-D. Ebbinghaus, et.al. [1990].