If you need to use a parametrization of the surface you can use the natural one

$\displaystyle \begin{pmatrix}x\\y\\4-x^2-y^2\end{pmatrix}$

$\displaystyle \overrightarrow{dS} = \frac{\partial}{\partial x}\begin{pmatrix}x\\y\\4-x^2-y^2\end{pmatrix} X \frac{\partial}{\partial y}\begin{pmatrix}x\\y\\4-x^2-y^2\end{pmatrix}\:dx\:dy$

$\displaystyle \overrightarrow{dS} = \begin{pmatrix}2x\\2y\\1\end{pmatrix}$

$\displaystyle \overrightarrow{F}.\overrightarrow{dS} = \left(2x^2y + 2y^2(4-x^2-y^2) + x(4-x^2-y^2)\right)\:dx\:dy$

which fortunately gives the same result as TheEmptySet