I am asked to evaluate the following integral:
$$ \iint_\Sigma(x^2+y^2)dS$$ where $\Sigma = \{(x,y,z) \in \mathbb{R}^3 : z = 2-(x^2+y^2) ,\, 0 \leq z \leq 2 \}$
To this end, I tried to use the parametrization $\phi_{u,v} = \begin{align} \begin{bmatrix} u\cos(v) \\ u\sin(v) \\ u \end{bmatrix} \end{align} $ with $u \in [0,\sqrt2], v \in [0,2\pi]$
The norm of the cross product is $\left\lVert\phi_u \times \phi_v\right\rVert = \sqrt2 u$.
I then get that the integral is equal to:
$$ \int_0^{2\pi}\int_0^{\sqrt2} u^2\sqrt2u \ du \ dv = 2\sqrt2\pi$$
However I am unsure, if this parametrization is valid, especially concerning the choice of $ v\in[0,2\pi]$.