Surface Integrals

Evaluate the following. (Give an exact answer.)
Question is
$\iint\limits_{S} f(x,y,z) \, dS$

f(x,y,z)= $\sqrt{x^2+y^2+z^2}$

$S: z=\sqrt{x^2+y^2}~,(x-1)^2+y^2\le1$

So after some work i arrive to to this.

$\sqrt{2}\int_0^2\int_{-\sqrt{1-(x-1)^2}}^{\sqrt{1-(x-1)^2}}\sqrt{x^2+y^2}\,dy\,dx$

which i then turn into sperical and get

$\sqrt{2}$ $\int_0^{\frac{2}{\pi}}\int_0^1~r^2\,dr\,d\theta$ which equals $2\pi\frac{\sqrt{2}}{3}$

But my book example has a similar question as

$2$ $\int_0^{\frac{2}{\pi}}\int_0^1~r^2\,dr\,d\theta$ which equals $\frac{4\pi}{3}$

Which is right? Why is theirs 2 and mine $\sqrt{2}$ and are my limits right? Or is it all wrong :( Thanks for the help.