Step 1

Given the force field,

$F(x,y,z)={x}^{2}i+2xj+{z}^{2}k$

Also, C is the surface boundary of S:$z=4-4{x}^{2}-{y}^{2},z\ge 0$

Then, C is an ellipse

$4{x}^{2}+{y}^{2}=4$

Then, using the Stoke's Theorem, work done W is given by,

$W={\int}_{C}F\cdot dr$

$=\int {\int}_{S}(\mathrm{\nabla}\times F)\cdot dS$...(1)

Here, $\mathrm{\nabla}\times F=(0,0,2)$

Also, note that the orientation of the surface is clockwise,

Hence,

$n=(0,0,-1)$

Step 2

Then, from equation(1),

$W=\int {\int}_{S}(0,0,2)\cdot (0,0,-1)dS$

$=-2\int {\int}_{S}dS$

$=-2\times (\pi \times 1\times 2)$

$=-12.566\text{}\text{units}$

Inverting the orientation of the surface S,

$W=\int {\int}_{S}(0,0,2)\cdot (0,0,1)dS$

$=-2\int {\int}_{S}dS$

$=-2\times (\pi \times 1\times 2)$

$=12.566\text{}\text{units}$

Thus, the correct option is,

b)Inverting the orientation of the surface S, we can apply Stoke's Theorem and conclude that $W=12.56$ units.