ABSTRACT
One of the fundamental problems in calculus is the computation of the area between the graph of a function f : [a, b]→ R and the x-axis. The essential ideas are illustrated in Figures 8.1 and 8.2. An interval [a, b] is divided into n subintervals [xk, xk+1], with a = x0 < x1 < · · · < xn = b. On each subinterval the area is approximated by the area of a rectangle, whose height is usually the value of the function f(tk) at some point tk ∈ [xk, xk+1]. In the left figure, the heights of the rectangles are given by the values f(xk), while on the right the heights are f(xk+1). In elementary treatments it is often assumed that each subinterval
has length (b−a)/n. One would then like to argue that the sum of the areas of the rectangles has a limit as n → ∞. This limiting value will be taken as the area, which is denoted by the integral
f(x) dx.
