Yup, that's the nicer way...In theory.
Use the divergence theorem to compute the flux over the closed hemisphere. Say you get f0 as your answer. Then compute the flux directly over the closed disk. This should be easy. Your g(x,y) = z = 0. Your normal vector will be <0,0,-1>.
But integrating that sin(x^3) will give problems, even if you switch to polar coordinates. Are you sure that's not a typo?