I have to evaluate this integral:
$$ \int_{0}^{\sqrt{2}}\int_{y}^{\sqrt{4-y^2}}\int_{0}^{\sqrt{4-x^2-y^2}} \sqrt{x^2+y^2+z^2}dzdxdy $$
in spherical coordinates. I see that the region in the xy plane is a circular sector bound by $y=x$ and $y=\sqrt2$ with a radius of 2, I have found that the region in three dimensions becomes complicated to evaluate because of the plane that cuts the spherical sector at y=sqrt(2). I am having trouble finding an expression for $\rho$ or r that describes both the spherical part and the planar part, as well as an $\phi$ that works as well, is see that $ \frac{\pi }{4}\leq \theta \leq \pi $.
Thanks