Direction of dS when integrating curl around a surface?

Question

I am attempting to solve a Stoke's Theorem question where I first find the curl of a vector field, and integrate it along a surface. In part b of the attached question I find the curl of the entire vector field but the solutions tell me that after taking the dot product with $dS$, only the $z$ component survives. Can someone explain how I can tell what $dS$ is and the direction it is in?

Answer 1

If the surface lies on the plane $z=1$, then its unit normal vector is $\vec{k}$, pointing in the positive $z$-direction by the right-hand rule and according to the sketch provided.