Surfaces/Areas of Revolution - Parametric

[KingBongo]

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.

 

[Office_Shredder]

Number two gives you the big hint

Do you have to impose some conditions, like df(t)/dt≥0 for all t?

What if df/dt<0 for some t? Then the x coordinate will double back on itself... but you don't know what happens with the y coordinate. So your curve isn't the curve of a function (since it's multi-valued). Try drawing some curves like that and see if anything gets messed up

 

[KingBongo]

Office_Shredder:
Yes, that is another possiblity, :) Mine was only an example. What I am looking for is what happens when the curve "turns" back in one or more of the coordinates and when it exists in more than one quadrants and stuff like that. Anything that can go wrong. That is why I am asking these questions.

 

[KingBongo]

Man, is this some kind of joke or what?

I am trying to find a parametric curve, x=f(t) and y=g(t), preferably lying in the first quadrant (x≥0, y≥0), fulfilling all of the objectives

A. Minimizing the surface of revolution around the x-axis, min S

B. Constant volume of revolution around the x-axis, V=const

C. End conditions f(a)=0, g(a)=R, g(b)=r, where R≥0, r≥0

In my mind this should be solvable, ending up with some kind of round curve (possibly part of a circle), but I am not able to find any feasible Extremals.

WHAT is wrong here? Does anybody have any idea? Are my formulas for the surface of revolution and volume of revolution correct?