Consider the bounded surface $S$ that is the union of $x^{2}+y^{2}=4$ for $-2 \leqslant z \leqslant 2$ and $(4-z)^{2}=x^{2}+y^{2}$ for $2 \leqslant z \leqslant 4$. Sketch the surface.

Using suitable parametrisations for the two parts of $S$, calculate the integral

$$\int_{S}(\nabla \times \mathbf{F}) \cdot d \mathbf{S}$$

for $\mathbf{F}=y z^{2} \mathbf{i}$.

Check your result using Stokes's Theorem.