I have a portion of a sphere defined by $x^{2}+y^{2}+z^{2}=4$, z ≥ 1, and want to change into spherical coordinates and compute the area of over the surface. For this I need to find the restrictions on the angles $\theta$ and $\varphi$. I know that the radius of the sphere is 2, and the the projection onto the xy plane gives a circle with radius $\sqrt3$.
I am having a hard time imaging in geometrically, how can I find the restrictions on the angles? Any help would be appreciated!
Given radius of the sphere is $2$, its surface can be parametrized in spherical coordinates as,
$x = 2 \cos \theta \sin\phi, y = 2 \sin \theta \sin \phi, z = 2 \cos \phi$
As we need to find surface area of the sphere for $z \geq 1$,
$\cos \phi \geq \frac{1}{2} \implies \phi \leq \frac{\pi}{3}$.
So limits of double integral to find surface area are $0 \leq \theta \leq 2\pi$ and $0 \leq \phi \leq \frac{\pi}{3}$.