Flux of a hemisphere

I have been asked to compute the flux through a hemisphere radius 2 centred at the origin oriented downward with $$\overrightarrow{F}=(y,-x,2z)$$ I have worked out that $$\hat{n}=-\frac{(x,y,z)}{2}$$ and in spherical co-ordinates, $$dS=4\sin(\gamma)d\gamma d\theta.$$ So the flux is given by $$\int_0^{2\pi}\int_0^{\frac{\pi}{2}}-2\cos(\gamma)4\sin(\gamma)d\gamma d\theta.$$

My question is, do I also need to calculate the flux through the bottom of the hemisphere, namely the disk $$x^{2}+y^2=4$$ on the $$xy$$ plane, or is this included in the integral above?

• It's included neither in the question nor in the integral you wrote. – Ted Shifrin Mar 5 at 18:29
• @TedShifrin how do you mean? Surely as the bottom part of the hemisphere is part of the surface it should be included in some part of the calculation? – Anon Mar 5 at 20:02
• Only if the instructions say to include the disk at the bottom of the hemisphere. – Ted Shifrin Mar 5 at 20:10
• Okay, thanks for clarifying – Anon Mar 5 at 20:16