The Integral as the Limit of a Sum of Rectangles

Abstract

Let f(x) be a bounded function on the finite interval [a, b]. The points with abscissae x₀, x₁, x₂,…,x_n−1, x_n where

$$a = {x_0} < {x_1} < {x_2} < \cdots < {x_{n - 1}} < {x_n} = b,$$

constitute a subdivision of this interval. Denote by m _v and M _v the greatest lower and the least upper bound, respectively, of f(x) on the v-th subinterval [x _{v −1}, x _v ], v= 1,2,…, n. We call

$$L = \sum\limits_{v = 1}^n {{m_v}} ({x_v} - {x_{v - 1}})\,{\text{the }}lower sum{\text{,}}$$

$$U = \sum\limits_{v = 1}^n {{M_v}} ({x_v} - {x_{v - 1}}){\text{the }}upper\,sum$$

belonging to the subdivision x₀, x₁, x₂,…, x_n−1, x _n . Any upper sum is always larger (not smaller) than any lower sum, regardless of the subdivision considered.