First, you need to solve for your axis of reference. For example, solve it as an expression of y. Then use double integrate but having a maximum and minimum value (Idk the english term) which is -R and R in both integrals if we are talking about the area of the entire sphere. Integrate normally with x and z to conclude.

The area is in picture. The fucking circle that is defined by crossing of a plane and a sphere. I don't know how to integrate this shit at all. I tried everything.

I'll think about it and get back to you. Btw, you need to find the area of the circle by integration?

Just calculate this integral? It's not as horrifying as I thought You take the below equations as given or did you calculate them yourself?

originally i need to and now i see that im a fucking idiot and that surface integral i posted is indeed an area.

Yeah, I knew I've missed it somewhere. I was gonna go back and check the calculations for errors and stuff I missed, but I got bored