parametrize surface region S is the elliptic region of the plane $y+z=1$ inside the cylinder $4x^2+4(y-0.5)^2=1$.
First parametrize $S$ using $(x,y,z)=G(u,v)$ and then calculate $\displaystyle \frac{dG}{du}\times \frac{dG}{dv}$.

How can I parametrize this crazy region??
Perhaps an additional question:  Find the centers of the Dandelin Spheres associated with this construction. Their points of tangency with the plane
are the foci of the ellipse. 

The parametrization : $ x = u/2 \cos(v) , y = 1/2 + u/2 \sin(v) ,  z = 1/2 - u/2 \sin(v)$   with  $ v =[0,2\pi)$ and $ u = [0,1]$ seems to work.
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\begin{align}
x&amp;\equiv \half\,\cos\pars{\theta}\,,\quad
y \equiv \half + \half\,\sin\pars{\theta}\,,\quad
z = 1 - y = \half - \half\,\sin\pars{\theta}\,;\qquad\qquad 0 \leq \theta &lt; 2\pi
\end{align}


  $$
\vec{\rm r}\pars{\theta}
=\half\braces{%
\cos\pars{\theta}\,\hat{x} + \bracks{1 + \sin\pars{\theta}}\,\hat{y}
+ \bracks{1 - \sin\pars{\theta}}\,\hat{z}}
$$