Extract from a possible on-line calculus textbook. Click on the second formula, or the arrowhead on the Y axis...
Definition: Let f(x) be a function defined on the [a,b] (closed) interval. Let us consider the following number:
where the xi series form an arbitrary partition of the [a,b] interval, and ξi is an arbitrary number in the interval [xi-1,xi]. This limit (if it exists) is called the definite integral or simply the integral of the function f(x) with respect to x over the interval [a,b] and is denoted as:
One possible interpretation of the definite integral is to represent the area enclosed by the curve of the function and the X axis. In this interpretation, a sum in the limit defining the integral forms an approximation of the area, which is refined when the sums tend toward their limit: