I've recently concocted an innocent looking problem that I'm struggling to solve myself.
The question is as follows: some wheel generates energy, in Joules, by spinning, at $W(\Theta)$ Joules per radian. The same wheel moves at an angular velocity $\omega (t)$, in radians per second. How many Joules are generated from $0 \le t \le 10$?
I've thought of multiplying $W(\Theta)$ and $\omega(t)$ because the units would cancel accordingly and allow for integration, but they are functions of different variables. I don't even have an answer because the problem is made up, but I am still curious. How would it be done? Is it even possible?
Since $W(\Theta)$ is in units of joules per radian and we want joules, we need only know through how many radians the wheel turns, then we can pass that angle through $W$ to get the number of joules.
We are told that the wheel turns at $\omega(t)$ radians per second. So over the interval $0 \leq t \leq 10$, the wheel turns through $$ \int_0^{10} \omega(t) \,\mathrm{d}t $$ radians.
Passing this angle to $W$, we compute $$ W \left( \int_0^{10} \omega(t) \,\mathrm{d}t \right) \text{,} $$ the number of joules generated during the interval $[0,10]$.