Riemann Integral - Overview

Overview

Let be a non-negative real-valued function of the interval, and let be the region of the plane under the graph of the function and above the interval (see the figure on the top right). We are interested in measuring the area of Once we have measured it, we will denote the area by:

The basic idea of the Riemann integral is to use very simple approximations for the area of By taking better and better approximations, we can say that "in the limit" we get exactly the area of under the curve.

Note that where ƒ can be both positive and negative, the definition of is modified so that the integral corresponds to the signed area under the graph of ƒ, that is, the area above the x-axis minus the area below the x-axis.