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How to parameterize solid of revolution?

  1. Mar 22, 2012 #1
    1. The problem statement, all variables and given/known data

    Here is the surface I need to parameterize. It is a solid of revolution.

    58wDQ.png



    2. Relevant equations



    3. The attempt at a solution

    So since its a piecewise function, I can define it as follows

    (x-2)^2 + z^2 = 1, 1<x<2
    z = -x+3, 2<x<3
    z = x-3, 2<x<3

    I know the formula for the area for a solid of revolution, but how do I parameterize this surface and then use that to calculate the surface area? I'm lost.
     
  2. jcsd
  3. Mar 23, 2012 #2

    LCKurtz

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    I'll show you how to parameterize one piece. That should get you started. I would express the circular arc like this:$$
    x=2+\cos\theta,\ z = \sin\theta,\ \frac \pi 2\le \theta\le\frac{3\pi} 2$$Now if you rotate that about the z axis, that will change the ##x## and ##y## values:$$
    x=(2+\cos\theta)\cos\alpha,\ y=(2+\cos\theta)\sin\alpha,\ z=\sin\theta$$Now if you let ##\alpha## very from ##0## to ##2\pi##, that will get the curved portion.

    That's the general method. But if you don't have to do it that way, a probably simpler method would be to just rotate the ds arc elements around.
     
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