Suppose I have a function $f(\theta)$ that is a function of the angle $\theta\in [0,2\pi)$.
Why is the average of $f$ over a large collection of randomly oriented objects:
$$\int f(\theta)\sin \theta\space d\theta ?$$
The $\sin{\theta}\space d\theta$ might be connected to the Jacobian for spherical coordinates? but I am not sure it makes sense to use that here?