Riemannsumma

Riemannsumma, more commonly known as the Riemann sum, is a method for approximating the definite integral of a function over a closed interval. Given a function f defined on [a,b], a partition P consists of points a = x0 < x1 < ... < xn = b, with subintervals [x_{i-1}, x_i] and widths Δx_i = x_i − x_{i-1}. Choosing a sample point x_i* within each subinterval, the Riemann sum is S(P, {x_i*}) = Σ_{i=1}^n f(x_i*) Δx_i. As the mesh (the maximum subinterval width) tends to zero, these sums may converge to the definite integral ∫_a^b f(x) dx.

If the limit exists and is independent of the choice of sample points, f is Riemann integrable on [a,b]. Functions that are continuous on [a,b] are guaranteed to be Riemann integrable, and the corresponding limit equals the integral. Riemann sums also relate to Darboux sums, with the same integrability criterion expressed through the convergence of upper and lower sums.

Common choices of sample points yield familiar rules: left endpoints, right endpoints, and midpoints lead to various approximations, while using endpoints of subintervals yields the trapezoidal rule. Riemann sums underpin numerical integration methods and provide a rigorous foundation for defining and computing the integral.

A simple example: the integral ∫_0^1 x dx can be approximated by left-endpoint sums with n subintervals, which approach 1/2 as n grows large, matching the exact value.