Surfaces/Areas of Revolution - Parametric

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Discussion Overview

The discussion revolves around the calculation of surfaces and volumes of revolution for parametric curves defined by x = f(t) and y = g(t) over a specified interval. Participants explore the conditions under which the formulas for surface area and volume are valid, particularly focusing on the behavior of the curve in relation to its quadrants and the implications of the derivatives of the parametric equations.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • Post 1 presents formulas for the surface area and volume of revolution and questions the conditions under which these formulas yield correct results.
  • Post 1 asks whether the curve must remain in a single quadrant and if conditions like df(t)/dt≥0 are necessary for valid calculations.
  • Post 2 highlights the implications of df/dt being negative, suggesting that this could lead to a multi-valued curve and complicate the surface and volume calculations.
  • Post 3 seeks clarification on the behavior of curves that turn back on themselves and exist in multiple quadrants, indicating a desire to understand potential pitfalls in the calculations.
  • Post 4 expresses a specific goal of finding a parametric curve that minimizes surface area while maintaining a constant volume, questioning the validity of the provided formulas and seeking guidance on feasible solutions.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the conditions necessary for the surface and volume formulas to be valid. There is no consensus on the implications of the curve's behavior, particularly concerning its quadrant positioning and the derivatives of the parametric equations.

Contextual Notes

Participants note potential complications arising from curves that double back on themselves or exist in multiple quadrants, but do not resolve these issues or provide definitive conditions for valid calculations.

KingBongo
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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.
 
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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
 
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.
 
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?
 

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