5. Let $S$ be the cylinder $x^2 + y^2 = 4$, between $z = 0$ and $z = 1$, oriented away from the $z$-axis. Let $C_1$ be the circle of radius 2 in the $xy$ plane, centered at the origin. Let $C_2$ be the circle of radius 2 in the plane $z = 1$, centered at $(0, 0, 1)$. Suppose $C_1$ and $C_2$ are both oriented counter-clockwise, as seen from $(0, 0, 10)$, looking down towards the origin. Suppose $\int_{C_1} \mathbf{F} \cdot d\mathbf{r} = 3$ and $\int_{C_2} \mathbf{F} \cdot d\mathbf{r} = \pi$. Calculate $\iint_S \text{curl } \mathbf{F} \cdot d\mathbf{A}$. You must draw a picture of this surface and the curves $C_1$ and $C_2$ to earn full credit!
