Given the frustum of a cone described by $z^2\ =\ x^2\ +\ y^2$ for $2\ \leq z\ \leq\ 8$, and the following generalized parameterization of a cone:
I would have thought the following parameterization in two variables would have been correct:
$r(u,v)\ =\ \langle \sqrt{2}/3 \cos u, \sqrt{2}/3 \sin u, v \rangle$
for $2 \leq v \leq 8$ and $0 \leq u \leq 2\pi$
The radius of the top circle of the frustum is $2 \sqrt{2}\ =\ a$ and the height of the frustum is $8\ -\ 2\ =\ 6\ =\ h$
So, $z\ =\ (rh)/a\ \Rightarrow r\ =\ (az)/h\ \Rightarrow\ r\ =\ \sqrt{2}/3$
And yet, my book tells me the correct parameterization is
$r(u,v)\ =\ \langle v \cos u, v \sin u, v \rangle$
for $2 \leq v \leq 8$ and $0 \leq u \leq 2\pi$
Doesn't the book's answer provide the wrong values for r at the top and bottom of the frustum? Am I misunderstanding something?
First of all, it is important to mention the parameterization of a surface (or a line) is not unique. There are usually different ways to describe a same surface, so just because you have a different answer as the one in the book does not necessarily mean that you (or the book) is wrong.
This being said, your parameterization is wrong. In fact, yours describes a cylinder (and not a cone). You should always check that when you plug the components of $r(u,v)$, they satisfy the cartesian equation of the surface. In your case, $$x^2(u,v)+y^2(u,v)=\frac{2}{9}\neq z^2(u,v).$$
You can check that the answer given by the book does satisfy $x^2(u,v)+y^2(u,v)= z^2(u,v)$, therefore it is correct.