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 using rectangles to approximate the area under the curve, trapezoids give a better approximation to the area.

The area of a trapezoid with base  and heights  and  is given by:

To approximate the area under the curve, add up the area of all the trapezoids.

which can be simplified to become the trapezoidal rule: