I was looking at a centrifuge, and I saw the following integral: $\int\omega^2\ dt$. I was wondering if this integral has any significance? Since this is a centrifuge, I would assume that $\omega$ refers to angular velocity, which has units $rad^2/s^2$. Integrating would give something with units $rad^2/s$, which doesn't seem to have to make any physical sense (it doesn't match the units for angular momentum, for example). So, what is the value of this integral?
Looks to me like a trademark. Yes, it doesn't quite make sense dimensionally, you have to multiply $\omega^2$ by the radius to get the acceleration. There is no such need to talk about radians$^2$ though: radians are pure numbers. In some reasonable model, the rate at which the centrifuge separates stuff out might be proportional to that acceleration. So integrate over time (with suitable constants) and you'd get the amount of stuff separated out.