(1) I have solved the problem, but I am not sure about the number of octants the surface covers (this affects the final answer value).
(2) Also, I have questions regarding the intersection of the two surfaces. On which plane does this intersection lie?
Intersection of the two surfaces is a unit circle: $x^2 + (y-1)^2 = 1$. On which
plane does this intersection lie? Not able to visualize this.
I considered the surface $S: z = \sqrt{x^2+y^2} = f(x,y)$. So, $dS = \sqrt{1 + f_x^2 + f_y^2} dxdy$
$dS = \sqrt2 dx dy.$
So the required surface area is $2\iint dS$. (Since the intersection of surface $S$ & the cylinder
covers 2 octants, while the intersection of the surface $z = – \sqrt{x^2+y^2}$
& the cylinder covers another two octants. Is this reasoning correct?)
So, final answer is $2\iint dS = 2\sqrt2 \iint dx dy = 2\sqrt2 \pi$.
Best Answer
Think about two circles attached at the origin. One opens toward above xy-plane, the other opens downward below xy-plane. The z value depends on the xy values when you traverse the circle.
You can change to cylindrical coordinate:
$$x=\cos{\theta}, y=\sin{\theta}+1$$
Then use either one of the equations to express $z$ in terms of $\theta$. That will be the parametric equation of the intersection plane.
I couldn't get a better picture. See below.