integral problem

Looking at the limits of integration, as well as the integrand, I think changing to spherical coordinates would make sense. In spherical coordinates, \(\displaystyle \sqrt{4- x^2- y^2- z^2}= \sqrt{4- \rho^2}\) and the differential of volume is \(\displaystyle \rho^2 sin(\theta)d\rho d\theta d\phi\). \(\displaystyle rho\) goes from 0 to 2, \(\displaystyle \theta\) goes from 0 to \(\displaystyle \pi\), and \(\displaystyle \phi\) goes from 0 to \(\displaystyle \pi/2\). The integral is \(\displaystyle \int_{\phi= 0}^{\frac{\pi}{2}}\int_{\theta= 0}^\pi \int_{\rho=0}^2 \sqrt{4- \rho^2} \rho^2 d\rho d\theta d\phi\).
 
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