I'm stuck in this exercise:
A solid is given which is the union of the cylindrical segment $$x^2+y^2 \le1,0 \le z \le2$$ and the half of the sphere $$x^2+y^2+(z-2)^2\le1 , z\ge2$$ Calculate the flux $$\iint_SF\cdot dS$$ where $$F(x,y,z) =(1,-1,z) $$ Initially, I thought of using Cylindrical coordinates to calculate the integral for the Cylinder and then use Spherical Coordinates to do the same for the sphere, but that doesn't seem right. Then, I thought of using the equations given to find the intersection equation but, honestly, I'm kinda stuck.
Now that you fixed the solid, I'm assuming that $S$ is its (closed) boundary; i.e. a cylindrical segment with half a sphere as a cap.
If you want to calculate the flux as a surface integral directly, you'll need to split it up in:
Because you're (probably) looking for the flux through a closed surface, I hope you know about the divergence theorem and if not, it's worth checking out. It states: $$\iint_SF\cdot dS = \iiint_G \nabla \cdot F \, dV$$ where $G$ is the given solid with boundary $S$.
The triple integral is easier to compute (in cylindrical coordinates) and in your case not even necessary, as $\nabla F = 1$; so the integral simplifies to finding the volume: $$\iint_SF\cdot dS = \iiint_G 1 \, dV = \mbox{Vol}\,(G)$$ And you can find that with basic formulas to avoid any integration at all.