You have been shown differentiation as a method for examining gradient, or even rate of turning, of a curve. More advanced use of differentiation looks at curvature in a more general sense. When you are first introduced to integration, you are almost always shown that it allows us to calculate the area underneath a curve. You are often shown integration as the opposite process to differentiation (which it is); since in mathematics once we have found a process of techniques we always almost immediately explore the reverse or inverse process as soon as we have understood the first (and sometimes even sooner). Integration, since it starts from a curve of gradient, will necessarily produce a family of curves and so an extra process, finding the constant of integration, is introduced.

I refer below to uses of integration, direct and indirect, only because that is the topic which is repeatedly the cause of the perennial “Is this any use in real life?”. Of course the matching question is “Is this in the exam?” (meaning quite patently that if it isn’t then they’re not going to listen) and I have been amused year after year that is is so often the very same student that asks both questions, seeing nothing strange, ambiguous or two-faced in their question.

What I must recommend when faced with an integration, from the advantage of 30 years of teaching, is a diagram. Such suggestion is almost always ignored at first. The reasoning is that a diagram, particularly drawing (sketching) the little strip, probably of length y and width dx, helps significantly in making you recognise what you are doing with the elements to be summed (integrated) and it does actually (double emphasis there) make you far more ready to accept new ways of using calculus. The few exceptions are on the occasions you are told ‘integrate this’; unless the integration is across a discontinuity, there is little point in establishing the shape of the curve (I’d wait until asking myself why the integration fails before doing a sketch in this circumstance); however, where any integration is constructed, such as a volume of revolution, the first example below, the sketch is a positive improvement, even when you’ve had loads of experience. Which is to say, I still draw a diagram, because it helps.

So, extensions of integration as a way of calculating the area under a curve would include:

a) taking the little strip and whirling it around an axis, so as to get a volume of revolution. If the axis of rotation is the x-axis the elemental disk has area πy² and thickness dx, so the integration ʃ πy²dx will produce a volume.

b) The same little strip based upon (beneath, on most diagrams) (x,y) has its centre of mass halfway up, at (x, y/2) so you can find the centre of mass of a defined shape by adding up those positions. In such cases, you are effectively doing moments and doing it in two directions, so the work can be significant, even after you used any symmetry to help reduce the work involved. In effect Mr = ∑mr where r is the position of an element of mass m, total mass M. The M may be found be integration and the integral of mass x position produces a result. The emboldened r is then the position of the centre of mass, which in turn is in effect the mean position, allowing us to replace an object with a model object of point mass.

c) Inertia is the second moment and simply squares the displacement MI= ∑mr² . The inertia expresses the unwillingness of a body to move. (I love the idea of social inertia, the resistance of a body of people to collective motion, which I claim is still the square of the number of people included). The calculus is easier than the idea it expresses. See the several pages in Mechanics about this. The calculation is a little easier if measuring the position relative to the centre of mass as the inertia is minimised when rotation goes through that point, but the maths allows you to calculate from anywhere.

d) Absolutely parallel to centre of mass in Mechanics is the idea of mean in Statistics. Add ’em up and divide by how many; for a continuous distribution (or one sufficiently numerous you can treat it as being continuous) you find the mean by ʃ ydx.

e) Equivalent to inertia in Mechanics is the idea of variance in Statistics. This computes the square of the deviation from the mean, absolutely the same idea as distance from the centre of mass in inertia. So if y=f(x) is a distribution function, then ʃ y²dx is hidden in the calculation of variance.

f) Funnily enough, the extension to ʃ y³dx occurs more readily in Stats than Mech and is a reflection of the wobble a distribution has from symmetry, called kurtosis. It is rarely discussed at A-level. Advanced Statisticians will pursue the fourth power integration too.

Very little of the list above does much more than gain you an A-level. Engineers want to know about centres of mass and inertia but their computer modelling package will include working out these values. Similarly, anyone doing heavy statistics will have all their data in a computer and their appropriate software will produce all sorts of statistical measures parallel to (and well beyond) the mechanical equivalents. So does that justify the inclusion of an understanding of calculus in school? Well, in the sense that you can’t go any further without it, yes, but for those whose maths will go no further, what was the point? Not much, I would agree. I used calculus quite a lot as a surveyor, but that was far more because I saw uses for the maths I knew than from any pressing need to use the material. I needed to use integration and numerical approximations to calculate the volumetric difference between surfaces – particularly the volume of tarmac, in two ways; what was supposed to be laid and what had been laid, since sometimes the client pays for what should be there and sometimes for what actually is there. I measured piles of sand by modelling; I approximated the cumulative frequency curve that is accumulated value with a cubic (rather than find an integral); I found several other uses of maths that weren’t calculus, mostly looking for simple linear relationships to describe things to others.

What I didn’t see then but do now is the wonderful field of differential equations. My favourite starting point is what I think of as ‘the growth curve’, dx/dt = Kx(1-x), where x is, say, height now and 1-x is growth remaining to occur and K is an obviously needed constant to scale matters appropriately. The integration requires Y13 Partial Fractions and produces ln(x/1-x)=kt+c, which in turn gives x = (1+Be^{-kt})^{-1} (you do it; it’s good practice) and B is usually found where t=0. This describes a whole lot of biology and other growth. It describes diseases, though the modelling of this rapidly becomes more complicated as you allow for the reduction in population because of death.