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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?

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    My guess is that this notion of averaging makes sense in the context of whatever source you encountered it in. Since you provided no context for the question, it is impossible to say whether this guess is true or not.2012-06-23

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