This section introduces a method to calculate area under a curve.

The Greek letter, Σ, is used to represent the sum of a series and is called sigma notation. It allows long sums (with many of terms) to be written in condensed form. In mathematics it is defined as following:

If a_m, a_m+1, ... , a_n-1, a_n are real numbers and m and n are integers such that m < n, then

The following formulas are some of the basic rules fro sigma notation in which c is a constant:

Area

To find the area under a curve of function y = f(x) between x = a and x = b, the interval can be equally divided into n pieces. The width of each piece is

     Δx = (b-a)/n

The left endpoint of these subintervals is

     x_i = a + iΔx

in which i = 0, 1, 2, ..., n-1.

These rectangles, whose height is f(x_i), are placed under the curve as shown on the left diagram. The area of each rectangle is f(x_i)Δx. Therefore, the area under curve y = f(x) is approximately equal to the sum area of these rectangles. It is

The more the rectangle used, the higher the accuracy is. Thus, the area can be expressed as the limit

Recall, previously the left edge top corner of the rectangular area was touching the curve. Now if the right side is used, the area can be expressed with

These two limits both will converge to the same value as n goes to infinite.

The Area Under Curve y = x²

This method can be better understood by finding the area under curve y = x² between 0 and 2.

First, equally divided the interval [0, 2] into n subintervals, the width of each subinterval is 2/n. The left x coordinates of the partitions are 0/n, 2/n, 4/n, ..., 2n/n. The height of each rectangle is (x_i)². The sum of the areas of the rectangles is

which can be rewritten as

Rearranging this expression gives

Taking the limit of this expression as n approaches +∞ gives the area.