TITLE: How to calculate this area?
QUESTION [8 upvotes]: Calculate the area of $$S:=\lbrace (x,y,z)\in\mathbb{R}^3: y^2+(z-x^2)^2=x^2-1\rbrace$$
Anyone have an idea? I tried using cylindrical coordinates, but nothing.
Well, I have something, but I'm not really sure... Ok:
We have the surface $$y^2+(z-x^2)^2=x^2-1 \Rightarrow (z-x^2)^2=x^2-1-y^2$$ 
$$\Rightarrow z-x^2=\sqrt{x^2-1-y^2} \Rightarrow z(x,y) =  \sqrt{x^2-1-y^2}+x^2$$
Now we put some reestriction on z. We have two cases: 
$$S^{-} = \lbrace C \cap \lbrace z<x^2\rbrace \rbrace$$
$$S^{+} = \lbrace C \cap \lbrace z>x^2\rbrace \rbrace$$
Let's work on $S^{+}$. Using the polar parametrization $x=rcos(\theta)$ e $y=rsin(\theta)$ we have:
$$z(\theta, r) = \sqrt{r^2cos(2\theta)-1}+r^2cos^2 (\theta)$$
Where we have 2 restrictions: $r>1$ and $r^2cos(\theta)>1$ (looking the root). So we can calculate the integration limits.
And we have the vector: $$\gamma(\theta, r) = (rcos(\theta), rsin(\theta), z(\theta, t))$$
So we only need use the area equation $$\int \int \left| \frac{\partial\gamma}{\partial \theta}\times \frac{\partial\gamma}{\partial r}\right|d\theta dr$$

REPLY [2 votes]: Reviewing the definition of an smooth surface again; we can form a Riemann Sum for a bounded function $f$ defined at all points of an smooth surface of finite area in $\mathbb R^3$. The area, as you noted above, seems to be infinite, so as @Greg said, I don't think the surface becomes finite.