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)

  • $\begingroup$ Like everywhere... $\endgroup$ – Hawk Sep 9 '13 at 2:48
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    $\begingroup$ @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. $\endgroup$ – Ryan Sep 9 '13 at 2:51
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    $\begingroup$ 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. $\endgroup$ – Evan Sep 9 '13 at 3:08
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    $\begingroup$ Analytic number theory. en.wikipedia.org/wiki/Analytic_number_theory $\endgroup$ – Eric Auld Sep 9 '13 at 3:27
  • $\begingroup$ 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." $\endgroup$ – Ryan Sep 9 '13 at 3:49

Ryan, perhaps a bit unexpected is the application of calculus in the human heart. More precise, cardiac output. The definition of cardiac output is the volume of blood pumped by the heart per unit time. The formula for this turns out to be a Riemann sum which in turn becomes an integral. And I find that unexpected in the sense that most people will look for calculus applications in physics/engineering or perhaps economics. But who generally thinks about calculus at work in our own hearts?

  • $\begingroup$ Also, influential applied mathematician from Oxford, Kevin Burrage, is currently modelling flow through the heart with fractional PDEs. $\endgroup$ – Bennett Gardiner Sep 13 '13 at 23:41
  • $\begingroup$ As an engineer myself, the heart is nothing but a pump and when analyzing something like blood flux I would expect calculus. As for biology: population problems, while involving discrete numbers, can be modeled by continuous calculus (but this is covered by rate of change). $\endgroup$ – Carlos Eugenio Thompson Pinzón Oct 16 '13 at 11:34

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