structural:reformatting, information:classification
Overview

Let f be a non-negative real -valued function on the interval, and let S be the region of the plane under the graph of the function f and above the interval. See the figure on the top right. This region can be expressed in set-builder notation as S = { (x, y): a ≤ x ≤ b, 0 < y < f (x) }. {\displaystyle S=\left\{(x,y)\,:\,a\leq x\leq b\,,\,0<y<f(x)\right\}.}

We are interested in measuring the area of S. Once we have measured it, we will denote the area in the usual way by ∫ a b f (x) d x. {\displaystyle \int _{a}^{b}f(x)\,dx.}

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

When f (x) can take negative values, the integral equals the signed area between the graph of f and the x -axis: that is, the area above the x -axis minus the area below the x -axis.