Let $E$ be the solid region defined by the inequalities

$x \ge 0$,

$0\le z \le \sqrt(x^2 + y^2)$,

$x^2 + y^2 + z^2 \le 4$

Suppose that $E$ has mass density $\mu(x,y,z) = xz$. Calculate the total mass of $E$.

I know how to set up the problem and how to do it, I get confused on determining the bounds. For example we can find:
$0 \le z^2 \le x^2 + y^2$ and then when $z=0$, we get the circle $x^2 + y^2 \le 4$.

I am confused how to find the bounds of the triple integral with this information.


If we visualize it, it's the reigon under a cone over that's in a sphere, like this: http://sketchtoy.com/67282842 (Sorry for the bad drawing skills) so I THINK this would be a parametrization $x=r\sin\phi \cos \theta, y=r\sin \phi \sin \theta, z=r\cos \phi, \phi\in [\pi/4,\pi], \theta\in [-\pi,\pi], r\in[0,2]$. Don't take my word for it, wait for other people to reply aswell!


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