The integral of a function  between the points  and  is denoted by

and can be roughly described as the area below the graph of  and above the -axis, minus any area above the graph and below the -axis, and all taken between the points  and .

The integral is important because it is an antiderivative for the original function, that is, if

then

    .

A Riemann sum is an approximation to the integral, that is, an approximation using rectangles to the area mentioned above. The line segment from  to  is split into  subsegments of equal width which form the bases of these rectangles, and the corresponding heights are determined by the value of  at some point  between the endpoints of the subsegment. The division of the segment  into subsegments is called a partition.

The Riemann Sum is given by the general formula:

There are five main types of Riemann Sums, depending on which point  is chosen to determine the height:

Instead of approximating the area under a curve using rectangles and trapezoids, parabolas can be used to approximate each part of a curve.

Simpson's rule is most accurate approximation using parabolas.

Let us compute the area under a parabola of the equation  passing through the three points :

As the points are on the parabola, we have:

Hence:

Therefore, the area under a parabola can be written as:

Hence by adding all the areas under each parabolic arc using three points we can derive:

The above equation can be simplified into Simpson's rule: