I want to calculate the flux of the vector field $$F(x,y,z)=(xz\sin(yz)+x^3)i+\cos(yz)j+(3xy^2-e^{x^2+y^2})k$$ through the surface $z=4-x^2-y^2$ for $z\ge0$ the orientation of the field is exterior.
So I applied the divergence theorem and calculated the following triple integral:$$\iiint_{\Omega}{z\sin{yz}+3x^2+z(-\sin(yz))dxdydz}$$. Now I tried to use spherical coordinates (http://mathworld.wolfram.com/SphericalCoordinates.html) but I get a very difficult integral. How am I supposed to do this problem? Thank you for you answer.