I have a surface given by $$x^2 + y^2 \le 4,\ x^2 + y^2 + z^2 \le 8\ \textrm{and}\ z\ge{-2}$$ and a force field given by $$\left<e^{cos{z}} + 3xy^2, \frac{1}{10-\sin{x}}+3x^2y, \sin{e^y}+z^3\right>$$
I need to find the outward flux through the surface. The surface looks like a cylinder with a dome top and a flat circular bottom. The divergence is just $$3y^2 + 3x^2 + 3z^2$$ and the flux integral (in cylindrical coordinates) becomes $$\int\limits_{0}^{2\pi}\int\limits_{0}^{2}\int\limits_{-2}^{\sqrt{8-r^2}}3(r^2+z^2)r\,dz\,dr\,d\theta$$ which can be evaluated.
The question is, do I need to subtract the flux of the bottom disk, or is it included? The integral of flux across that would be $$\int\limits_{x^2+y^2\le4}{-8 + \sin{e^y}}\,dx\,dy$$ which I wouldn't know how to solve.
