i.10.3 Flux of the vector field

Condition: Find the flow of the vector field \( \vec{f} \) through the surface (S) directly by definition: \[ \begin{array}{l} \vec{a}=(1+\sqrt{z}) i+(4 y-\sqrt{x}) i+x y k \\ S:\left\{\begin{array}{c} z^{2}=4\left(x^{2}+y^{2}\right) \\ z=3, x \geq 0 \end{array}\right. \end{array}\]