Referring to Figure 11, suppose that $$\oint_{{\mathcal{C}}_2} \mathbf{F} \cdot d\mathbf{s} = 3\pi,\qquad \oint_{{\mathcal{C}}_3} \mathbf{F} \cdot d\mathbf{s} = 6\pi$$ Use Green's Theorem to determine the circulation of $\mathbf{F}$ around ${\mathcal{C}}_1$, assuming that curl$_z(\mathbf{F})=3$ on the shaded region.

$\int_{{\mathcal{C}}_1} \mathbf{F} \cdot d\mathbf{s} =$