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 nth 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 nth 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.