Surface Integration Problem

1. The problem statement, all variables and given/known data
Let $D=\{(x,y,z)| z^{2}=1+x^{2}+y^{2} , 1<z<3\}$ Compute$\int\int_{D}zdS$

2. Relevant equations
From lectures I know;
$\int\int_{D}\delta dS=\int\int_{D}\delta\sqrt{(\frac{\partial f}{\partial x})^{2}+(\frac{\partial f}{\partial y})^{2}+1}dxdy$

3. The attempt at a solution
I'm not sure what I'm doing is correct my answer seems wrong;

$z^{2}=1+x^{2}+y^{2}$
so

$z=\sqrt{1+x^{2}+y^{2}}$

Taking partial derivatives of x and y and substituting into equation from 2 I get

$\int\int_{D}\sqrt{5x^{2}+5y^{2}+1}dxdy$

making change of variables to cylindrical coordinates

$\int\int_{D}\sqrt{5r^{2}+1}r drdt$

making substitution

$u=5r^{2}+1$

I get

$\int\int_{D}\sqrt{u} dudt$

I brought the limits through as well to have limits for u of 0 and 41 and limits of t of 0 and 2$\pi$

giving me a final answer of 1099.675108 which seems completely wrong;

Any suggestions?
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 I got 2 instead of 5. Really, $\frac {\partial z}{\partial x} = \frac{x}{z}$ and $\frac {\partial z}{\partial y} = \frac{y}{z}$ $z \sqrt{1+\frac{x^2}{z^2} + \frac{y^2}{z^2}} = z \sqrt{ \frac{z^2 + x^2 + y^2}{z^2} } = \sqrt{1+2x^2+2y^2}$ I then got 73.39 which is a lot more likely.

 Tags calculus, integration, surface