Definition – Integral – Math-reference.net

Contents

1. Introduction
2. Definitions

Definition – Riemann Integral

Let $f$ be a bounded function defined on $[a, b]$ , and let $X = {x_0, ..., x_n}$ be a partition of $[a, b]$ . Finally, let

$\begin{array}{ll} U(f, P) = \sum\limits_{i=1}^n M_i(x_i-x_{i-1}) \\\\ L(f, P) = \sum\limits_{i=1}^n m_i(x_i-x_{i-1}) \end{array}$

be the upper and lower sums respectively. Then, define,

$U(f) = inf\{U(f, P)~|~\text{P is a partition of}~[a, b]\}$

$L(f) = sup\{L(f, P)~|~\text{P is a partition of}~[a, b]\}$ .

If $U(f) = L(f)$ , then we say that the function $f$ is Riemann integrable on $[a,b]$ .

We assume that $f$ must be bounded as this implies the existence of $U(f)$ and $L(f)$ .