Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$

Short Answer

Evaluate $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ , where $\mathbf { F } ( x , y , z ) = x ^ { 2 } \mathbf { i } + x y \mathbf { j } + z \mathbf { k }$ and S is the part of the surface $z = x ^ { 2 } + y ^ { 2 }$ below the plane $z = 1$ , with upward orientation.