Correct Parametrization for Calculating Area of a Tube?

[Redwaves]

Homework Statement

Write the parameterization of a tube $r = \frac{1}{2}$ around C then find the area of this tube:

$C \subset R^3$ = Circle of radius 1 at the origin in the plane xy

Relevant Equations

$S(s, \theta) = \gamma (s) + r \beta (\theta)$

Hi,

I'm trying to find the area of this tube using $\int \int ||\vec{N}|| ds d\theta$. However, I get 0 as result which is wrong.

So at this point, I'm wondering if I made a mistake during the parametrization of the tube. This is how I parametrized the tube.
$S(s, \theta) = (cos(s), sin(s) , 0) + \frac{1}{2} cos(\theta)\vec{N(s)} + \frac{1}{2} sin(\theta)\vec{B(s)}$
= $S(s, \theta) = (cos(s), sin(s) , 0) + \frac{1}{2} cos(\theta)(-cos (s), -sin (s), 0) + \frac{1}{2} sin(\theta)(0,0,1)$

 

[pasmith]

Your parametrization appears to be correct, although your definition of \theta is not how I would define it: In cylindrical polars (r,s,z) you're taking the circle (r - 1)^2 + z^2 = \frac14 and revolving it about the z axis. So first I would set r = 1 + \frac12\cos\theta and z = \frac12 \sin \theta and then I would set x = r(\theta) \cos s and y = r(\theta)\sin s.

Please show the rest of your working: What do you get for \|\vec N\|, and what limits are you using for your surface integral?

 

[Redwaves]

I get $\vec{N} = (-cos(s), -sin(s) , 0)$

And the limits I'm using are $[0,2\pi]$ for both ds and $d\theta$, since I have a circle moving around a circle.