Applications of Calculus II to the real world A lot of my calc II students are asking me what are the real world applications of what we are studying in Calc II (right now we are studying methods of integrations, so of course one of the applications is in finding areas and volumes, are there any other cool applications? I mean something that can be explained in a simple way to a calc II student). Later we will study series and sequences. I'm just looking for ways to pick up the interest of my students, do you have any ideas?
 A: Fourier Analysis is a very useful range of study based on the idea of definite integration. It is, in fact, the very foundation of digital signal processing. The discrete Fourier transform is used to process digital signals everywhere. You use it in your iPod, your laptop, and your television. It is defined by:
$$\hat f(\xi) = \int_{-\infty}^{\infty} f(x) e^{2 \pi ix \xi} dx$$
Where $\xi$ is a frequency in Hertz.
A: Its hard to find real world applications of calculus II which are not in a book already. Here are some interesting applications:


*

*Explain how you can write circular motion in vector form as $r(t) = (\cos(\omega t), \sin(\omega t))$. Then it is straight forward to derive the centripetal force using calculus - this is usually not derived in high school physics.

*Talk about springs and simple harmonic motion and its differential equation. Show the solution. Define kinetic and potential energy and show what they are for simple harmonic motion. Show that the total energy is conserved by showing the derivative of the sum of kinetic and potential energy is zero.

*Give examples of some hard and not obvious optimization problems which are not in the book. Talk about the brachistochrone problem and its solution which is a cycloid. They won't be able to solve it but you can tell them that it is solved using higher level calculus (specifically calculus of variations). It is also used to reformulate all of mechanics using Lagrangians.
Sorry I didn't find cooler applications. But here are some interesting applications/theorems/examples within mathematics:


*

*Define the Gamma function. Use integration by parts to show that it is the same as a factorial for integer values.

*Prove that if $f'' + f = 0$, then $f(x) = a\sin(x) + b\cos(x)$ (I read a really nice proof in Spivak's Calculus). This is also the differential equation of simple harmonic motion which I mentioned above.

*Prove that $e$ is irrational using its series expansion (its actually a short proof and not hard to follow).

*Prove using Rolle's theorem that if $f' = 0$ for all $x$, then $f$ is a constant function - this is usually taken for granted without proof.

*Show that a function can have a derivative at only one point: $f(x) = x^2$ for rational $x$ and $-x^2$ for irrational $x$. Then $f'(0) = 0$

*Show that a derivative of a function can be discontinuous: $f(x) = x^2 \sin(1/x)$. Then $f'$ is not continuous at $0$.

*Show that some functions are not derivatives of any other function: $f(x) = 0$ for all $x$ except $f(0) = 1$. Then $f$ cannot be a derivative.
A: As has been said elsewhere, there are loads of applications to classical mechanics and electromagnetism (both statics and dynamics). tylerc0816 mentioned statistics, which is great, but an awful lot of that looks like black magic without more advanced knowledge (that I don't have yet).
