I'm trying to determine the area of the 2 surfaces that is formed when the paraboloid $z=x^2+y^2$ splits the sphere $x^2+y^2+z^2=1$.
So I thought that i I first can determine the surface area of the sphere by the parameterisation
$$ r(s,t) = (R \sin(t) \cos(s) , R \sin(t) \sin(s) , R \cos(t)) $$
and therefore the area is
$$ \text{Area(sphere)} = ..... = \int_0^{\pi}\int_0^{2\pi} R^2 \sin(t) dt ds = ... = 4\pi . $$
But I don't really know how I can determine the surface area of the upper part! (for lower part I thought that I'll subtract). Any tips?
From your two equations, you can find the intersection: $$x^2+y^2+z^2-1=0\\z^2+z-1=0$$ Only one solution ($z_+$) is in the $[-1,1]$ interval.
You already started using spherical coordinates, so all you need is to find $s_+=\arccos(z_+)$, and change the limit of integration for $s$ from $s_+$ to $\pi$ or from $0$ to $s_+$, depending on which surface you choose.