Mapping a cylinder onto a sphere

[c299792458]

Homework Statement

How might I show that the map (in cylindrical polar coordinates) given by $f:(1,\phi,z)\to(\sqrt{1-z^2},\phi,z)$ does not change the area?

Homework Equations

The Attempt at a Solution

I can see this is like having a sphere in a cylinder and we shine "light" on the cylinder orthogonal to its axis inwards towards its axis so that the image falls on the unit sphere housed inside. A hint to the problem is given to be that the area of the spherical disc of spherical radius $r$ is $2\pi(1-\cos r)$.

 

[c299792458]

Don't worry, I have solved it, thanks for reading though.