Definition: The Definite Integral.

If $f(x)$ is a function defined on an interval $[a, b]$, the definite integral of $f$ from $a$ to $b$ is given by

$$
\int_a^b f(x) d x=\lim {n \rightarrow \infty} \sum_{i=1}^n f\left(x_i^*\right) \Delta x
$$

provided the limit exists. If this limit exists, the function $f(x)$ is said to be integrable on $[a, b]$, or is an integrable function.

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