"Calculus: Alternate Fifth edition: Larson/Hostetler/Edwards"
Chapter 12 section two pg. 692
If a smooth curve C given by and does not cross itself on an interval , then the area S of the surface of revolution fomred by C about the coordinate axes is given by the following:
Revolution about x-axis g(t)>0
and Revolution about y-axis f(t)>0
This is the best calculus book I have ever used