Let And Let S Be the Boundary Surface of the Solid

Multiple Choice

Let $\mathbf { F } ( x , y , z ) = \left( x ^ { 3 } + y \sin z \right) \mathbf { i } + \left( y ^ { 3 } + z ^ { 2 } \sin z \right) \mathbf { j } + \left( z ^ { 3 } + x \right) \mathbf { k }$ and let S be the boundary surface of the solid E bounded by $z = \sqrt { 4 - x ^ { 2 } - y ^ { 2 } } , z = \sqrt { 1 - x ^ { 2 } - y ^ { 2 } }$ , and $z = 0$ . Evaluate the surface integral $\iint _ { S } \mathbf { F } \cdot d \mathbf { S }$ .

A) $\frac { 62 \pi } { 5 }$

B) $\frac { \pi } { 2 }$
C) $\pi$
D) $\frac { 4 \pi } { 3 }$
E) $\frac { 2 \pi } { 3 }$
F) $\frac { 192 \pi } { 5 }$

G) $\frac { 186 \pi } { 5 }$

H) $4 \pi$