I have to find the parameterization of the surface that is part of the cylinder $$y^{2}=2-x$$ for $x\geq0$, bounded by the cylinders: $$y^{2}=z\quad\text{and}\quad y=z^{3}$$ for $0\leq y\leq1$.
I could draw the surface and I can see it. I tried to write $x$ and $z$ as functions of $y$ (or $x$ and $y$ using $z$), but for me is, for example: $$z=y^{2}$$ and $$z=\sqrt[3]{y}$$ How could I work with it? Which one should I use? (Sorry for the English)
The surface is $y^2 = 2-x. $ The rest of it establishes the boundaries of the surface.
One way to do it would be:
$x = 2-y^2\\ y = y\\ z = z$
With $0\le y \le 1$ and $y^2 <z<\sqrt [3] y$