(1) The problem may require you to use the Stoke's Theorem, as you have indicated in the title "stocks question";
(2) Use the Stoke's Theorem reduces the problem to 1D;
(3) Solving the problem in 2D is very complicated, sometimes even impossible.
OK. Let's try direct surface integration step by step, instead of using the Stoke's theorem and see how more complicated it is.
(Note: the image you provided indicate the third component is , rather than . Anyway, it does not influence the final answer)
Taking advantage of the center symmetry,