Calculate the area of $\partial G$ of $G = (z > x^2 + y^2$ and $x^2 + y^2 + z^2 < z)$ in $\mathbb{R}^3$.

Question

Calculate the area of $\partial G$ of $G = (z > x^2 + y^2$ and $x^2 + y^2 + z^2 < z)$ in $\mathbb{R}^3$.

I'm not really sure how to approach this. I've tried using spherical coordinates but I didn't really get anywhere. I know that I have to integrate over some manifold, but I'm not sure how to look at it. Any help would be appreciated!

Answer 1

The domain G is axisymmetric. So you can let $r^2=x^2+y^2$ and plot the domain with polar coordinate system. The two domains looks like below. As you can see, the intersection of the two domains is null. So the area is zero.