The r is the radius, so if it's around the y axis, the radius may be x, and if it's around the x axis, the radius could be f(x). Once again, if y = f(x), f'(t) is 1, so it just becomes rsqrt(1 + f'(x)).
It's also important to note that this is very similar to the shell formula. Here, this is done using frustums, so instead of many hollow cylinders, there are many thin, hollow frustums.
In polar, arc length is different, so the formula above would change.
If you wanted to derive it, you could say y = f(θ)sin(θ) and x = f(θ)cos(θ)