A surface is created when the cylinder y^2 + z^2 = 1 intersects the cylinder x^2 + z^2 = 1. Find the area of this surface.

I need some help setting up this problem.
I think the formula i use is:

A(S) = sqrt(1 + (dz/dx)^2 + (dz/dy)^2) dA

Now i'm not sure what dz/dx and dz/dy would be. I believe i would first have to solve for z giving:
z= sqrt(1-y^2)
z= sqrt(1-x^2)

But how/where do i get dz/dx and dz/dy from? Do i just take dz/dx and dz/dy from the two equations above?