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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".

Figure %: Computing the Area of R(f, a, b)

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].

Figure %: Splitting up the interval [a, b] into n smaller intervals

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).

Figure %: Containing R(f, a, b) with rectangles

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).

Figure %: Rectangles 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

f (x)dx    

This symbol above, and the number it represents, are also referred to as the definite integral of f (x) from a to b.

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