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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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    If $f$ is the constant function 17, then the average of $f$ is 17, but the integral is zero, so it would appear that you do not have a question.2012-06-23
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    @GerryMyerson: THanks, I think the average should just be ${1\over 2\pi}\int f(\theta)\,\,\,d\theta$ or if there is some sensible pdf, but clearly $\sin$ is not a valid pdf, so I really don;t understand what that is doing! O well...2012-06-23
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    How is the angle $\theta$ distributed?2012-06-23
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    @ncmathsadist: I would think uniformly, since the objects are "randomly distributed"?2012-06-23
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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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