Integration is an important topic discussed in first year calculus. Many subsequent topics, such as solving first-order or second-order differential equations, requires the evaluation of at least one integral, and sometimes several.

The most challenging material in integration is arguably the techniques of integration. Unfortunately, due to time constraints, only basic techniques are commonly taught, while in reality, many integrals having special forms are solvable only by using more sophisticated techniques.

A surveyor often encounters problems like Problem 1:

Determine the length of side AC of the right triangle ABC with the right angle at B. A lake intersecting the sides AC and CE prevents the direct measurement of their sides.

As I write (Spring 2012) I am aware that I have been a member of the Mathematical Association for fifty years. I joined during the annual conference in 1962 when Dr Combridge was the main figure in the society. The highlight was a session at the conference (at King's College, London) by Geoffrey Matthews showing how matrices might be introduced into the school curriculum. Some treated this as hilarious, others as a signpost to the future. My friend Frank Budden and I regarded it as a premature ploy, which, we thought, might not tum out to be such a good idea. We subsequently wrote a successful book Mathematics through Geometry (1964) arguing in detail why the modem mathematics revolution—which was then gathering pace—might end in tears. We identified spatial imagination as the heartland of mathematics, and contrasted it with a denial of spatial imagination in the abstract calculi which were all the vogue at the time.

A pleasurable aspect of mathematics and its teaching is to review the diversity of ways in which theorems are proved. Especially in elementary branches, there are various kinds of proof: using (or avoiding) spatial geometry, analytic or coordinate geometry, common algebra, vectors, abstract algebras, matrices, determinants, the differential and integral calculus, and maybe mixtures thereof. Further, sometimes a proof of one kind is elegant while another is clumsy, or one proof of a theorem suggests why it follows while another proof is not perspicuous. There is also the question of whether a proof is direct or indirect (for example, proofby contradiction).