In the same vein as your bicycle with a speedometer, we present some more situations and questions below. These are not questions with exact answers, nor even the types of questions asked in secondary-school mathematics; but they raise issues about how you might approach them mathematically. How might they also require us to add something up over many short intervals to get close to the exact answer — that is, by using integration? Some of them are contrived and unrealistic, but perhaps they will be useful to think about.
- Today is partly cloudy. Sometimes the sun is shining fully, sometimes the sun is partially blocked by clouds, and sometimes the clouds are thicker or thinner. You look at the sun and you know how much light is getting through at every instant. How hot is it? Do you need to look down to see if the ground is still wet from the rain last night? Will you get sunburnt?
- You are in control of a spacecraft heading towards Mars at 21 000 kilometres per hour. To land the spacecraft safely you need to slow down to zero. The spacecraft has rockets, and you know, when the rockets are turned on, the force they exert on the spacecraft. How long do you need to fire the rockets to stop the spacecraft?
- You are observing an ecological system in the forest. There are predators and prey. The number of predators increases when there is prey for them to eat, and the more prey around, the faster their numbers increase. The number of prey increases naturally but decreases when they are eaten by predators. How will the ecosystem evolve over time?
- You are at a power station and you can see, at every instant, the power that the station is producing. How much energy has been produced over the course of the day?
- You are lucky enough to have a smartphone with an accelerometer in it. That is, the phone can tell its acceleration at any instant. By using only its accelerometer, can the phone calculate which way it is going at any instant and how fast? Can it know at any instant where it is?
- You have a function \(f(x)\) and its graph \(y=f(x)\). What is the area under the graph?
It is the last problem — an area under a graph — which is a purely mathematical one, and the most abstract. However, all of the other situations, it turns out, can be related to this type of mathematical problem. Integration is intimately connected to the area under a graph.