If $S=\{x^2+y^2=z,z\leq2\}$ is oriented upwards and $F=(P,Q,R)$ is a $C^2$ field s.t. $Q_x=8,P_y=5$, find $\iint_S$ curl $F$

Question

I interpret that this exercise wants us to use both Stokes' and Green's Theorems (The field is $C^2$ and both the surface and its boundary are smooth). Hence,

$$\iint_S \text{curl} \textbf{ F }dS \stackrel{\text{Stokes}}{=} \oint_{\partial S} \textbf{F } ds \stackrel{\text{Green}}{=} \iint_D Q_x -P_y \ dA = 3 \big( \text{Area of }D \big) = 6\pi$$

Is this correct?