Babylonian astronomers tracked Jupiter with a rule computers still use for areas
Long before calculus, Babylonian sky-watchers used the trapezoidal rule to work out how far Jupiter moved along the ecliptic. The same idea, adding up weighted samples of a function to estimate the area beneath it, now sits at the heart of numerical integration, the family of algorithms for computing definite integrals approximately.
Numerical integration, also called quadrature, estimates the value of a definite integral to a chosen accuracy; some authors reserve cubature for more than one dimension. There are good reasons not to find an exact antiderivative. The function may be known only at sampled points, as in some embedded systems. Its antiderivative may not exist in elementary form, as with e to the minus x squared, whose integral is the error function. Or the exact answer may be an infinite series or a special function that is harder to evaluate than a direct estimate.
Most methods evaluate the function at a set of integration points and combine the results as a weighted sum, with the points and weights chosen by the method and the accuracy needed. A good method keeps the error small with few evaluations, which saves time and cuts accumulated arithmetic error. The crudest approach is a Riemann sum over tiny steps, which works for any piecewise continuous function of bounded variation.
Quadrature originally meant constructing, with compass and straightedge alone, a square equal in area to a given figure, a Pythagorean way of thinking about area. Rectangles, parallelograms and triangles yield to such constructions, and the lune of Hippocrates can be squared, but squaring the circle was proved impossible in the 19th century. Archimedes, using Eudoxus's method of exhaustion, showed that a sphere's surface is four times its great circle and that a parabolic segment has four-thirds the area of its inscribed triangle, the high point of ancient analysis.
Medieval and early modern mathematicians favoured the less rigorous but more powerful method of indivisibles. Galileo and Gilles de Roberval found the area under a cycloid arch, and Grégoire de Saint-Vincent studied the hyperbola in 1647, with his pupil Alphonse Antonio de Sarasa linking that area to logarithms and so revealing the natural logarithm. John Wallis wrote what are now definite integrals in 1656. The phrase numerical integration first appeared in 1915, in a course book by David Gibb.
Source: Numerical integration
