$$\iiint_D \,dz\,dy\,dx$$
where the region $D$ is defined as followed:
$$0<z<\sqrt{9-x^2-y^2}$$ $$0<y<\sqrt{9-x^2}$$ $$0<x<3$$
I got the corresponding spherical coordinates for $D$:
$$0<\rho<3$$$$0<\theta<\pi/2$$ $$0<\phi<\pi/2$$ where $\theta$ is the angle between $x$ and $y$, $\phi$is the angle between $z$ and $xy$ plane.
Have I correctly calculated this region of integration in polar coordinates?
First cartesian condition: upper half-sphere(center=$O$, radius=3).
Second cartesian condition: half-cilynder(axis=$Z$, radius=3). As sphere $\subset$ cilynder, only the $0<y$ inequality is relevant.
Third cartesian condition: slice. As sphere $\subset$ ($x<3$) only the $0<x$ inequality is relevant.
Bottom line: you are cutting the sphere by the three positive half-spaces $x>0$, $y>0$, $z>0$.