Riemannszummá

Riemannszummá, commonly called a Riemann sum, is a method for approximating the value of a definite integral by partitioning a closed interval [a, b], evaluating the integrand at a chosen point in each subinterval, and summing the products of the function values with the subinterval widths. A Riemann sum is written as S(P, f, x_i*) = sum_{i=1}^n f(x_i*) Δx_i, where Δx_i = x_i − x_{i−1} for a partition P: a = x_0 < x_1 < ... < x_n = b.

The definite integral is defined as the limit of Riemann sums as the mesh size max Δx_i tends to zero, provided the limit exists and is independent of the choice of sample points x_i*. If this limit exists, f is Riemann integrable on [a, b], and its value is the Riemann integral. Continuous functions on [a, b] are always Riemann integrable; more generally, a function with a set of discontinuities of measure zero is integrable. Functions with extensive oscillations, such as the Dirichlet function, may fail to be Riemann integrable.

In practice, one may use left, right, or midpoint sums as specific choices of sample points, and compare upper and lower Darboux sums, which use the supremum and infimum of f on each subinterval. The Riemann integral connects to the fundamental theorem of calculus: if f is integrable and F its antiderivative, then ∫_a^b f = F(b) − F(a). The concept underpins numerical integration and the rigorous treatment of area and accumulation.

Historically, the idea was developed by Bernhard Riemann in the mid-19th century as a foundation for a rigorous theory of integration and analysis.