Riemann integral

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of theintegral of a function on an interval.^[1] For many functions and practical applications, the Riemann integral can be evaluated by using the fundamental theorem of calculus or (approximately) by numerical integration.

The Riemann integral is unsuitable for many theoretical purposes. Some of the technical deficiencies in Riemann integration can be remedied with theRiemann–Stieltjes integral, and most disappear with the Lebesgue integral

The integral as the area of a region under a curve.

A sequence of Riemann sums over a regular partition of an interval. The number on top is the total area of the rectangles, which converges to the integral of the function.

The partition does not need to be regular, as shown here. The approximation works as long as the width of each subdivision tends to zero.