Evaluating 3D Integral in Rectangular Coordinates

[Ions]

Problem: Evaluate (leave in rectangular coordinates):
<br /> <br /> \int_{-1}^{{1}}}\int_{-{\sqrt{1-x^2}}}^{{\sqrt{1-x^2}}}\int_{-{\sqrt{1-x^2-y^2}}}^{{\sqrt{1-x^2-y^2}}}\ \,dz\,dy\,dx<br />

 

[n!kofeyn]

Well, start slow. Can you evaluate this integral?
<br /> \int_{-{\sqrt{1-x^2-y^2}}}^{{\sqrt{1-x^2-y^2}}}\ \,dz<br />

 

[Ions]

<br /> <br /> 2\int_{-1}^{{1}}}\int_{-{\sqrt{1-x^2}}}^{{\sqrt{1-x^2}}}\2{\sqrt{1-x^2-y^2}}}\,dy\,dx<br />

Supposedly it's easier to make a subtitution u = 1-x^2 at this point but I don't see how...