I think we have to use the Stoke's Theorem here.
So let $F=(-y^3+xz)i+(yz+x^3)j+(z^2)k$. Then Curl $F=-yi+xj+3(x^2+y^2)k$. Now $\int \int_S Curl F.n dS=\int_C F.dr=$The integral we have to compute. But I am confused how $S$ looks like. How do we find $n$ and $dS$? Can somebody please help me?