555-704i Riemann Integral Notes Spring 2007

1. The Darboux definition of the Riemann Integral Let f : [a, b]→ R be a it bounded function. Definition 1.1. A partition P of [a, b] is collection of points {x0, · · · , xn} which satisfiy a = x0 < x1 < · · · < xn = b. Denote by ∆xk the length of the interval [xk−1, xk], i.e., ∆xk = xk−xk−1. Then the mesh |P| is by definition equal to max1≤k≤n∆xk. Given a bounded function f and a partition P of [a, b] we define the Riemann upper sum by