The Definite Integral

If $f$ is defined on $[a,b]$ and the limit

$$\lim_{\left\lVert\Delta \right\rVert \to 0} \sum_{i=1}^n f(c_i) \Delta x_i$$

exists, then $f$ is integrable on $[a,b]$ and the limit is called the definite integral of $f$ from $a$ to $b,$ and is denoted by

$$\lim_{\left\lVert\Delta \right\rVert \to 0} \sum_{i=1}^n f(c_i) \Delta x_i = \int_a^b f(x) dx.$$