In the previous section we saw how to evaluate the approximate area of a region bounded by the graph of a positive function, the X-axis and two vertical straight lines, by using a series of rectangles to help us calculate the area we are looking for. There are two types of approximations: a greater approximation (above the curve) which we called the upper sum (S_n) and a lower approximation (below the curve) called the lower sum (s_n) such that if A is the area we are looking for then:

where n and m are any two natural numbers. Also, in the previous example for a certain value of n, the difference between the upper sum and the lower sum S_n-s_n tends to zero when n tends to infinity. This is not generally the case for any function, but always the case when the function we start with is continuous over the interval where the area is being calculated.

Let's form a general rule from the above situation. Let's take a continuous function over a closed interval [a,b], where it is irrelevant as to whether or not it is positive over the interval. Answer the following questions referring to the graph below:

Follow the method outlined in the previous section to find the area bounded by the graph of the function, the X-axis and the straight lines x=-2 and x=3 to one decimal place. What is the result like? Is it an acceptable result for an area?

The result of the exercise above shows us that the method does not always give the correct result for the area of a plane figure. However, it also shows us that if the difference between the upper sum and the lower sum tends to zero then both sums tends to a certain number when n tends to infinity. In this case we can say that the function can be integrated on the interval [a,b] and that the number we obtain is called the

integral of the function f(x) on the interval  [a,b].

This number is denoted by .

When the function f(x) is positive over the interval [a,b] (as was the case in the previous section) the integral coincides with the area bounded by the graph of the function, the X-axis and the straight lines x=a and x=b. If f(x) is not positive then this is not the case.