I am trying to calculate the area of the intersection between a spherical segment and a spherical cap as a function of the angle of the spherical segment with respect to the sphere's equator. The spherical cap's area is defined as follows (from Wiki):
$A = 2 \pi r^2(1-\cos\theta)$
where $\theta$ is the polar angle between the rays from the center of the sphere to the apex of the cap and the edge of the disk forming the base of the cap.
So far, I have been approaching the problem as an integral over the latitudinal angle $\phi$:
$\int_0^\phi 2 \theta(\phi) \rho(\phi) dh$
where
$\theta(\phi) = \theta_0 \cos(\phi \frac{\pi}{2 \theta_0})$,
$\rho(\phi) = r \cos(\phi)$, and
$dh = r d\phi$.
So essentially, I am finding the partial circumference across the spherical cap as a function of the latitudinal angle and integrating over the desired range of latitudinal angles.
Calculating this integral and comparing the result of the limit as $\phi \to \theta_0 $ to the equation for the area of a spherical cap I see that the results do not match up and my result always ends up slightly smaller. I am not sure exactly what my issue is, but I think it might be related to how I am calculating the partial circumference at increasing latitudinal angles. Thanks for the help.