Calculus: Concepts and Methods

This chapter is chiefly about integrating real valued functions of one real variable. Because an integral is a generalisation of the idea of a sum, it is also convenient to include some discussion of summation in this chapter. It is assumed that readers have some previous knowledge of these subjects. In particular, §10.2– §10.9 inclusive consist of an accelerated account of elementary integration theory which a newcomer to the topic would find difficult to assimilate adequately unless they had an unusual aptitude for mathematics. However, experience shows that students are often very rusty on the techniques of integration and it is strongly advised that Exercises 10.10 be used as a check on how well the reader recalls the relevant material before moving on to more advanced work. It is also suggested that all readers study §10.4 with some care. The notation for indefinite integrals can be very confusing if imperfectly understood.

The material given in §10.11– §10.12 about infinite series and integrals over infinite ranges will probably be new to most readers of this book. It is not essential for most of what follows and is best omitted if found at all difficult to understand. The same applies to §10.14 on power series. This is not to say that these are unimportant subjects: only that it may be advisable to leave their study until a later date. There remains §10.13 on differentiating integrals. Although this will be new, the technique can be very useful in some contexts.