So in Spherical coordinates the integrand is

$\displaystyle \sqrt{x^2+y^2+z^2}=\sqrt{\rho^2}=\rho$ why don't we need an absolute value?

Now Sketch the region of integration and you will see the it is the top half of a sphere of radius 1.

Now to trace a sphere is radius 1 $\displaystyle \rho \in [0,1]$

$\displaystyle \theta \in \left[0,\frac{\pi}{2}\right]$

$\displaystyle \phi \in [0,\2pi]$

Don't forget to change the differential of volume to spherical coordinates as well.

$\displaystyle dzdydx=\rho^2\sin(\theta)d\rho d\theta d\phi$