RiemannIntegration

RiemannIntegration, commonly known as Riemann integration, is the standard method of defining the definite integral of a function on a closed interval by means of limits of Riemann sums. Given a partition P of [a,b], with a = x0 < x1 < ... < xn = b, and a choice of sample point ξi in each subinterval [xi-1, xi], the Riemann sum is S(f, P, ξ) = sum f(ξi)(xi − xi-1). If the limit of these sums exists as the mesh of the partition tends to zero and is independent of the choice of ξi, then f is Riemann integrable on [a,b], and the limit is the Riemann integral ∫_a^b f(x) dx. The upper and lower sums, U(f, P) and L(f, P), use sup and inf of f on each subinterval, and integrability holds when the limits of these sums converge to the same value.

Sufficient conditions for integrability include that f is continuous on [a,b], and more generally that the set of discontinuities of f on [a,b] has measure zero. Classic examples illustrate the scope: the Dirichlet function (1 on rationals, 0 on irrationals) is not Riemann integrable, while Thomae’s function (zero at irrationals, 1/q at rationals p/q in lowest terms) is integrable with integral 0.

Key properties of Riemann integration include linearity, additivity over adjacent intervals, and compatibility with limits of sequences of functions under suitable conditions. The Fundamental Theorem of Calculus links Riemann integration with differentiation for functions that are differentiable in a suitable sense. Improper Riemann integrals extend the notion to unbounded intervals or unbounded integrands via limits. Numerical methods such as the trapezoidal and Simpson rules provide practical approximations of the integral. Riemann integration precedes and contrasts with Lebesgue integration, which generalizes the concept further.