I am struggling on this problem.

I am asked to evaluate

$\oint \vec r \dot \,d\vec \sigma$

over the whole surface of the cylinder bounded by $x^2+y^2=1, z=0, z=3$

It seems pretty straight forward geometrically as it is just a unit circle at $z=0$ and then it extends up to $z=3$ forming a cylinder.

Initially, I parametrized the unit circle as $\langle cos(t), sin(t)\rangle$ and

$d\vec \sigma$ pointing normal in the positive z direction.

This did not go anywhere however. Now I am thinking I should just work in cylindrical coordinates, but I am just getting kind of confused working with this. Does anyone have any tips for me?

Thank you always!