Suppose that you have a parametric curve given by x = f(t), y = g(t), a ≤ t ≤ b What will the Surface of revolution and Volume of revolution around the x-axis be? I have two candidates: Surface: S = 2*pi*int( |g(t)|*sqrt( (df(t)/dt)^2+(dg(t)/dt)^2 ) , t=a..b) Volume: V = pi*int( (g(t))^2*df(t)/dt , t=a..b) I believe those are correct, at least I hope so... Now, here come my main questions; 1. Under what circumstances will you get "correct" results? Under what circumstances does the surface/volume "overlap" when rotating? 2. Does the curve have to behave in some certain way? Does it have to stay in one quadrant for all t? Do you have to impose some conditions, like df(t)/dt≥0 for all t? Could somebody please explain this to me? I have been trying to figuring that out.