Evaluate using spherical coordinates: $\displaystyle \int \int \int dx dy dz$

$\displaystyle T: 0 \le x \le 1 \; \; ; \; \; 0 \le y \le \sqrt{1-x^2} \; \; ;\; \; \sqrt{x^2 + y^2} \le z \le \sqrt{2 - (x^2 + y^2)}$I can represent this in cartesian coordinates like this:

$\displaystyle \int_0^1 \int_0^{\sqrt{1-x^2}} \int_{\sqrt{x^2+y^2}}^{\sqrt{2-(x^2+y^2)}}dzdydx$

and from here convert to spherical coordinates. I'm not sure how to completely convert this to spherical coordinates.

The only thing I could see was that on the xy-plane, part of the area is a quarter of a circle in the first quadrant with radius 1 (since y goes from 0 to $\displaystyle \sqrt{1-x^2}$ and x goes from 0 to 1), so $\displaystyle \theta$ would go from 0 to $\displaystyle \frac{\pi}{2}$.