We will develop the definite integral as a means to calculate the area of certain regions in the plane. Given two real numbers a < b and a function f (x) defined on the interval [a, b], define the region R(f, a, b) to be the set of points (x, y) in the plane with a≤x≤b and with y between 0 and f (x). Note that this region may lie above the x-axis, or below, or both, depending on whether f (x) is positive or negative. In computing the area of R(f, a, b), it will be convenient to count the regions above the x-axis as having "positive area", and those below as having "negative area".
We can split up the interval [a, b] into n smaller intervals (for some integer n) of width Δx = (b - a)/n. Let
|si = a + i(Δx)|
for i = 0, 1, , n, so that the n intervals are given by [s0, s1], ,[sn-1, sn].
Let Mi be the maximum value of f (x) on the interval [si-1, si]. Similarly, let mi be the minimum value of f (x) on the interval [si-1, si]. Consider the region made up of n rectangles, where the i-th rectangle is bounded horizontally by si-1 and si and vertically by 0 and Mi. As shown below, this region contains R(f, a, b).
Moreover, we know how to compute the area of this region. It is simply
|(M1) + (M2) + ... + (Mn) = Mi|
We denote this nth upper Riemann sum by Un(f, a, b). Replacing Mi in the above with mi, we obtain a region contained in R(f, a, b).
The area of this region is equal to
|(m1) + (m2) + ... + (mn) = mi|
called the nth lower Riemann sum and denoted by Ln(f, a, b). Recall that in computing these sums, we are counting areas below the x-axis as negative.
For nicely behaved functions, Un(f, a, b) and Ln(f, a, b) will approach the same value as n approaches infinity. If this is the case, f is said to integrable from a to b. The value approached by both Un(f, a, b) and Ln(f, a, b) is what we call the area of R(f, a, b) and is denoted by
This symbol above, and the number it represents, are also referred to as the definite integral of f (x) from a to b.