Correct Parametrization for Calculating Area of a Tube?

Redwaves
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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)##
 
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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?
 
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.
 
There are two things I don't understand about this problem. First, when finding the nth root of a number, there should in theory be n solutions. However, the formula produces n+1 roots. Here is how. The first root is simply ##\left(r\right)^{\left(\frac{1}{n}\right)}##. Then you multiply this first root by n additional expressions given by the formula, as you go through k=0,1,...n-1. So you end up with n+1 roots, which cannot be correct. Let me illustrate what I mean. For this...
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