Evaluate $\int _ { S } \vec { F } \cdot \vec { d A }$

Essay

Evaluate $\int _ { S } \vec { F } \cdot \vec { d A }$ , where ${ \vec { F } } = x \vec { i } + 6 y \vec { j }$ and S is the part of the cylinder $x ^ { 2 } + y ^ { 2 } = 4$ with x $\ge$ 0, y $\le$ 0, 0 $\ge$ z $\le$ 2, oriented toward the z-axis.