I have not grasped the way to solve these kinds of problems yet. I need to parametrize the surface and find its area: $S:x^2+y^2+z^2=4$ with $z \ge\frac{\sqrt{x^2+y^2}}{3}$.
I have already parametrized the surface of the sphere this way:
$S:(x,y,z)=(2\sin{\phi}\cos{\theta},2\sin{\phi}\sin{\theta}, 2\cos{\phi})$
But how do I find the limits of integration to find the area of the part of the sphere enclosed by $z \ge\frac{\sqrt{x^2+y^2}}{3}$?
$z \ge\frac{\sqrt{x^2+y^2}}{3}\\ 2\cos\phi \ge\frac{2\sin\phi}{3}\\ 3 \ge\tan\phi\\ \phi \le\tan^{-1} 3$