The height, in centimeters, of a bicycle pedal is given by $h(t)=30+16\sin t$ where $t$ is the time. Evaluate and interpret the following integral \begin{align} \dfrac{1}{2\pi}\int_0^{2\pi} h(t)\,dt. \end{align}
The answer is $30$. Not sure about the interpretation. Had it been speed against time I suppose the area under the graph is the vertical displacement. Divided by $2\pi$ might be the average vertical displacement per time. But I am stuck on this.
Well, first of all let's take a look at the calculation. We have:
$\frac{1}{2\pi}\int_0^{2\pi}h(t)dt=\frac{1}{2\pi}\int_0^{2\pi}30dt+\frac{1}{2\pi}\int_0^{2\pi}16\sin(t)=\frac{30\cdot 2\pi}{2\pi}+0=30$
Integrating over $[0,2\pi]$ simply means the pedal going through one through circle. Obviously the height has to be the same after that. So, dividing by $2\pi$ hasn't that much physical meaning but is rather used for scaling.
Note: It seems a little bit odd though that you scale the second summand as well... That doesn't really make sense to me.