Define the Gauss map of a smooth embedded surface. Consider the surface of revolution $S$ with points

$$\left(\begin{array}{c} (2+\cos v) \cos u \\ (2+\cos v) \sin u \\ \sin v \end{array}\right) \in \mathbb{R}^{3}$$

for $u, v \in[0,2 \pi]$. Let $f$ be the Gauss map of $S$. Describe $f$ on the $\{y=0\}$ cross-section of $S$, and use this to write down an explicit formula for $f$.

Let $U$ be the upper hemisphere of the 2 -sphere $S^{2}$, and $K$ the Gauss curvature of $S$. Calculate $\int_{f^{-1}(U)} K d A$.