Let $\mathbf { F } ( x , y , z ) = \left( x ^ { 2 } + y e ^ { x } \right) \mathbf { i } + \left( y ^ { 2 } + z e ^ { x } \right) \mathbf { j } + \left( z ^ { 2 } + x e ^ { y } \right) \mathbf { k }$ and let S be the boundary surface of the solid $E = \left\{ ( x , y , z ) \mid x ^ { 2 } + y ^ { 2 } 1,0 \leq z \leq x + 2 \right\}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

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The Answer of Let $\mathbf { F } ( x , y , z ) = \left( x ^ { 2 } + y e ^ { x } \right) \mathbf { i } + \left( y ^ { 2 } + z e ^ { x } \right) \mathbf { j } + \left( z ^ { 2 } + x e ^ { y } \right) \mathbf { k }$ and let S be the boundary surface of the solid $E = \left\{ ( x , y , z ) \mid x ^ { 2 } + y ^ { 2 } 1,0 \leq z \leq x + 2 \right\}$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

Let And Let S Be the Boundary Surface of the Solid