Volume of a Parametrised Surface

  • #1

Homework Statement


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 .


Homework Equations




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.
 

Answers and Replies

  • #2
Dick
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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.
 
  • #3
LCKurtz
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Here's a picture if that helps you. In the picture ##0\le t \le 5##.
picture2.jpg
 

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  • #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:
 
  • #5
Dick
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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:
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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