After a morning exploring the Indiana Dunes National Lakeshore, we headed north through Chicago and into Wisconsin, where we stopped to tour the Jelly Belly factory. I have to say that as factory tours go, this one was pretty disappointing. However, there were three good things about it: (1) it was free; (2) we got free samples of jelly beans; and (3) I found some math.
Fundamentally, calculus can be boiled down to doing two things: making a distance as small as possible (differential calculus) or adding up a bunch of little things (integral calculus). If the little things that we are adding up all represent areas, then the sum of those things is a volume. This seems inherently strange to students, so we try to use examples and analogies to help them understand what’s happening. For example, imagine that you have a bunch of DVDs lying around on your floor. If you look at each individual one, it’s pretty thin, but if you stack them up all of a sudden you have a right circular cylinder, which more obviously has a volume. That’s the basis for finding volumes through calculus.
We call this the method of disks or method of washers, and an analogy like this usually makes sense to students pretty quickly. The other method for finding volumes is called the method of cylindrical shells, and it’s a little harder to explain with an example, or at least it was until now.
When they make the jelly beans, they form the cores by machine. They then put those cores in a large rotating drum, similar to the kind you might see a mason use to mix mortar. As the cores rotate in the drum, a person alternates between ladling liquid syrup and very fine sugar into it. Each core picks up a thin layer of syrup, which the sugar helps to solidify. Each new layer on the jelly bean is very thin, but by the time they’re finished the bean has more than doubled in size from its original core.
Like before, we start with something that’s very thin, and if we put enough of the thin things together then we get a volume. The math is a little different because we’re doing it with thin shells instead of thin disks, but the result is the same, and now I have a way to explain it.