I try to calculate the intensity of a reflected signal from an ensamble of planar objects, randomly oriented in space.
With respect to the incoming beam, the orientation of the object is defined by two angles, $\phi$ and $\theta$, as sketched here: https://cloud.ill.fr/index.php/s/JjhSHxY3glwhyAY
For any given orientation, the signal $f(\phi, \theta)$ is as follows:
$$f(\phi, \theta) = \left\{ \begin{array}{l} 1 \text{ for } \theta<\theta_c \text{ and any } \phi \\ 0 \text{ for } \theta>\theta_c \text{ and any } \phi \end{array} \right.$$
In particular, I am interested in the result of the orientational average over $\phi$ (which I note as $\left<f(\phi, \theta)\right>$ and then calculating the limit for $\theta\rightarrow0$.
Edit: I have clarified the question better following the comment of David.
As I do not want to integrate over $\theta$, is the correct form of the average integral still $\int_0^{2\pi} f(\phi, \theta) d\phi$? i.e., the first part of: $$ \int_0^{2\pi}d\phi\int_0^{\pi} f(\phi, \theta)\sin\theta d\theta $$
thanks and best wishes, Leonardo