Consider the vector field $\mathbf{v}=(x^2y-\sqrt[3]{z}, \sqrt[3]{z}-xy^2,z)$ and a surface given by $\frac{z^2}{2}=x^2+y^2$ and $z\in[0,2]$. Calculate the flux of $\mathbf{v}$ through the surface.
I did calculate the divergence of $\mathbf{v}$ to using the divergence theorem , but I dont know how to write the limits of the integral.
Hint: Use cylindrical coordinates $z, \rho, \varphi$:
$$\sqrt{2}\rho \le z \le 2, \quad 0 \le \rho \le \sqrt{2}, \quad 0 \le \varphi \le 2\pi.$$