Unexpected Practical Applications of Calculus

Calculus shows up in a lot of places in the world. Specifically, here are three areas where I see it used the most:

1. Optimization problems.
2. Anything involving rates of change (e.g. velocity $\rightarrow$ acceleration).
3. Anything involving "averages" (e.g. surface area).

I am more interested in the non-intuitive and unexpected applications of Calculus, however. For instance, the Fourier Transform is an alright example. But in some ways I still feel like Calculus isn't totally unexpected here, as it becomes really intuitive once you understand that the integral is just computing the average power at each signal frequency.

So, in what fields/areas of science does Calculus pop up unexpectedly? Preferably those applications which are practical in the real world. (i.e. not number theory)

• Like everywhere... – Hawk Sep 9 '13 at 2:48
• @sidht Any places that don't directly involve optimization, rates of change, or averages? If you can think of some, then please do post an answer. – Ryan Sep 9 '13 at 2:51
• I don't think Fourier transform counts as an unexpected application, since the integral is involved. You might ask where calculus /isn't/ involved. Off the top of my head, anything discrete math won't use calculus (immediately), and I think this is where you might find your example, because often we take continuous approximations to discrete objects (limiting behaviors as a problem size gets larger) to give us information about the original discrete object. – Evan Sep 9 '13 at 3:08
• Analytic number theory. en.wikipedia.org/wiki/Analytic_number_theory – Eric Auld Sep 9 '13 at 3:27
• The title to this question originally said "Unexpected Practical Applications of Calculus." I removed the 'practical' because it seemed a bit wordy. I think I'll add it back though, because there are a lot of trivial answers to this question along the lines of "number theory." – Ryan Sep 9 '13 at 3:49