Riemannintegrálhatóként

The Riemann integral is a formal way to assign a number to the area under a curve on a closed interval. Let f be a bounded function on [a, b]. A partition P divides the interval into subintervals by points a = x0 < x1 < ... < xn = b. For each subinterval, choose a sample point ti in [xi-1, xi], and form the Riemann sum S(P, t) = sum f(ti)(xi − xi-1). The function is Riemann integrable on [a, b] if the limit of S(P, t) exists as the mesh size (the maximum subinterval length) tends to zero, and this limit is denoted ∫ from a to b f(x) dx.

A basic and widely used criterion is that every continuous function on [a, b] is Riemann integrable. More generally, a bounded function with only a finite or even a countable set of discontinuities of measure zero is Riemann integrable (the Lebesgue criterion). Classic counterexamples include the Dirichlet function, which is 1 on rationals and 0 on irrationals in [0, 1]; this function is not Riemann integrable because its upper and lower sums do not converge to a common value.

The integral relates to antiderivatives through the Fundamental Theorem of Calculus: if f is integrable on [a, b] and F is an antiderivative of f, then ∫_a^b f(x) dx = F(b) − F(a). Conversely, if F is differentiable with F' = f on (a, b), then ∫_a^b f(x) dx = F(b) − F(a).

Extensions include improper Riemann integrals, which handle unbounded intervals or unbounded integrands via limits. The Riemann integral is a foundational notion in analysis and a precursor to the more general Lebesgue integral.