As an example for the computation of the area of a surface of rotation with surface integrals, my textbook derives the formula for the area of a sphere of radius $R$.
The sphere is described by $x^2+y^2+z^2=R^2$. In order to be able to use $Area=2\pi\int\limits_{a}^{b}{r\sqrt{1+\phi'(r)^2dr}}$, we parametrize the top half of the sphere like so: $z=\phi(x)=\sqrt{R^2-x^2}$ $(0\leq{R})$, before rotating it around the $z$-axis.
How did we get this parametrization? Where did $y^2$ disappear to?