information:extraction
Connection with integration

For a one-dimensional Riemann sum over domain {\displaystyle }, as the maximum size of a subinterval shrinks to zero (that is the limit of the norm of the subintervals goes to zero), some functions will have all Riemann sums converge to the same value. This limiting value, if it exists, is defined as the definite Riemann integral of the function over the domain, ∫ a b f (x) d x = lim ‖ Δ x ‖ → 0 ∑ i = 1 n f (x i ∗) Δ x i. {\displaystyle \int _{a}^{b}f(x)\,dx=\lim _{\|\Delta x\|\rightarrow 0}\sum _{i=1}^{n}f(x_{i}^{*})\,\Delta x_{i}.}

For a finite-sized domain, if the maximum size of a subinterval shrinks to zero, this implies the number of subinterval goes to infinity. For finite partitions, Riemann sums are always approximations to the limiting value and this approximation gets better as the partition gets finer. The following animations help demonstrate how increasing the number of subintervals (while lowering the maximum subinterval size) better approximates the "area" under the curve:

Since the red function here is assumed to be a smooth function, all three Riemann sums will converge to the same value as the number of subintervals goes to infinity.