Net flux through a closed sphere

[PeteyCoco]

Find the net electrical flux through a closed sphere of radius R in a uniform electric field

I know that the flux is going to be 0 since there is no charge enclosed, but how would I show this mathematically? The next half of the question asks about a cylinder with sides parallel to the electric field, which I can prove is 0 easily, but I'm not sure if I know the math to prove the first scenario. Can the sphere-problem be proven with only knowledge of Single-Variable calc?

EDIT: I guess I'm asking if this can be proven easily using the first half of Gauss's Law, ignoring (Q-internal)/(epsilon-nought)

 

[Redbelly98]

Yes, it can be done with a single-variable integral.

You can visualize that the sphere has an "axis" aligned with the electric field. For area elements, take rings that are centered on that axis, something like this:

Integrate the flux over all the area elements in a hemisphere, and you'll get that hemisphere's contribution to the total flux.

Hope that's clear enough.