Parametric
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.
Polar
In polar, arc length is different, so the formula above would change.
R is f(θ)sin(θ) if you're rotating about the polar axis (θ = 0). and f(θ)cos(θ) if you're rotating about the line θ = pi/2
If you wanted to derive it, you could say y = f(θ)sin(θ) and x = f(θ)cos(θ)
David Witten