Evaluate $$\int_C ydx+z^2dy+xdz$$ on a specific curve, the intersection between $z=\sqrt{x^2+y^2}$ and $z=6-(x^2+y^2)$.
I don't know how to parametrize this, also it seems wrong to me if I subtitute $z^2$ in the second one i get the equation $z=6-z^2\implies z=2$ since $z\geq0$. what do I do with this..? Also is there some kind of way to always approach these questions where you have to integrate between 2 surfaces?
You are on the right track. $z=2$ tells you that the curve $C$ is contained in the plane $z=2$. Moreover $z=6−(x^2+y^2)$ (or $z=\sqrt{x^2+y^2}$) implies that $x^2+y^2=6-2=2^2$. What kind of curve is this? It should be easy to get a parametrization now and evaluate $$\int_C ydx+z^2dy+xdz.$$