We now begin three chapters about integration.

As the word implies, integration is about bringing things together. In mathematics the process of integration involves bringing together function values in a special way to form a summation. This sum of function values is called an integral and we shall see that there are many circumstances in which we calculate the value of some quantity by the process of integration. At first sight it will appear that integration is just the opposite of differentiation, i.e. in a sense what we appear to be doing is ‘undoing’ the differential of a function. This process is called anti-differentiation. For example, cos (x) is the derivative of sin (x), and sin (x) is the anti-derivative of cos (x).

However, integration and antidifferentiation are conceptually different. The fact that the two processes give the same answer is a consequence of an important theorem called the Fundamental Theorem of Calculus. In this first chapter we define what we mean by the integral of a function and investigate the results of integrating the basic functions we have met before.