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
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 .