Volume of a Parametrised Surface

[Tom31415926535]

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.

 

[Dick]

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

 

[LCKurtz]

Here's a picture if that helps you. In the picture ##0\le t \le 5##.

 

[Tom31415926535]

Thanks heaps! That’s very helpful

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

 

[Dick]

Thanks heaps! That’s very helpful

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

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