RIEMANN INTEGRAL

The Riemann integral is defined in terms of Riemann sums of functions with respect to tagged partitions of an interval. Let [a,b] be a closed interval of the real line; then a tagged partition of [a,b] is a finite sequence

Riemann sums converging as intervals halve, whether sampled at ■ right, ■ minimum, ■ maximum, or ■ left.

This partitions the interval [a,b] into n sub-intervals [x_i−1, x_i] indexed by i, each of which is “tagged” with a distinguished point t_i ∈ [x_i−1, x_i]. A Riemann sum of a function f with respect to such a tagged partition is defined as

thus each term of the sum is the area of a rectangle with height equal to the function value at the distinguished point of the given sub-interval, and width the same as the sub-interval width. Let Δ_i = x_i−x_i−1 be the width of sub-interval i; then the mesh of such a tagged partition is the width of the largest sub-interval formed by the partition, max_i=1…n Δ_i. The Riemann integral of a function f over the interval [a,b] is equal to S if:

For all ε > 0 there exists δ > 0 such that, for any tagged partition [a,b] with mesh less than δ, we have

<img src="http://upload.wikimedia.org/math/5/b/c/5bcae45e90b157b48189d638bdb0d244.png&quot; alt="\left| S – \sum_{i=1}^{n} f(t_i)\Delta_i \right|

When the chosen tags give the maximum (respectively, minimum) value of each interval, the Riemann sum becomes an upper (respectively, lower) Darboux sum, suggesting the close connection between the Riemann integral and the Darboux integral.