Summation and the Definite Integral

Calculating the area of a two dimensional object or the volume of a three dimensional object has been a subject of study for mathematicians and natural philosophers since antiquity. When the enclosed area or volume is a rectangle or a cube the calculations are simple enough. However, when the enclosed area is irregularly shaped our only option is to break the area up into smaller areas with shapes we can easily calculate, and then sum those smaller areas (or volumes) together to reach an estimate of the area under consideration. If we break the subject area into infinitely many infinitesimally small regions we can reach exact values for the area or volume under consideration. This process of summing an infinite number of infinitely small areas is called integration.

The definite and indefinite integrals

There are two types of integrals based on the results they produce. One is called the definite integral and it produces a number, a value that can be characterized as the area under a curve. The definite integral is identified by the upper and lower limits associated with the integral symbol. These limits serve as boundaries on the area we wish to calculate.