Definition of the Definite Integral
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
|s i = a + i(Δx)|
for i = 0, 1, , n , so that the n intervals are given by [s 0, s 1], ,[s n-1, s n] .
Let M i be the maximum value of f (x) on the interval [s i-1, s i] . Similarly, let m i be the minimum value of f (x) on the interval [s i-1, s i] . Consider the region made up of n rectangles, where the i -th rectangle is bounded horizontally by s i-1 and s i and vertically by 0 and M i . As shown below, this region contains R(f, a, b) .
Moreover, we know how to compute the area of this region. It is simply
|(M 1) + (M 2) + ... + (M n) = M i|
We denote this n th upper Riemann sum by U n(f, a, b) . Replacing M i in the above with m i , we obtain a region contained in R(f, a, b) .
The area of this region is equal to
|(m 1) + (m 2) + ... + (m n) = m i|
called the n th lower Riemann sum and denoted by L n(f, a, b) . Recall that in computing these sums, we are counting areas below the x -axis as negative.
For nicely behaved functions, U n(f, a, b) and L n(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 U n(f, a, b) and L n(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 .