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Volume of a Parametrised Surface

  1. Oct 13, 2018 #1
    1. The problem statement, all variables and given/known data
    Let C be the parametrised surface given by

    Φ(t,θ)=(cosθ/cosht, sinθ/cosht,t−tanht), for 0≤t and 0≤θ<2π

    Let V be the region in R3 between the plane z = 0 and the surface C.

    Compute the volume of the region V .

    2. Relevant equations

    3. The attempt at a solution
    I thought I needed to perform a change of variables; changing from x,y,z to theta,t,z.

    I tried to find the Jacobean for this, but it came to zero. I'm pretty sure it was wrong in the first place, but I have no idea what to do otherwise. Your assistance would be greatly appreciated.
  2. jcsd
  3. Oct 13, 2018 #2


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    It's a surface of revolution. To visualize what it looks like, figure out what cross sections of constant ##t## are.
  4. Oct 13, 2018 #3


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    Here's a picture if that helps you. In the picture ##0\le t \le 5##.
  5. Oct 14, 2018 #4
    Thanks heaps! That’s very helpful :smile:

    I’m a little uncertain about how to setup the integral to calculate the volume though. Am I correct in needing to perform a change of variables? If so, what am I doing incorrectly that produces a Jacobean of zero?

    Thanks :smile:
  6. Oct 14, 2018 #5


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    You don't do that sort of a change of variables. You have only two parameters so you have a two dimensional surface. In the three dimensions ##x, y, z## that will have zero volume. That's why your Jacobean is zero. The problem is actually easier than that. As I said you can treat it as a solid of revolution. https://en.wikipedia.org/wiki/Solid_of_revolution
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