Answer
$ \int_{0}^{\pi/2}\int_{0}^{3} \int_0^{2}f(r \cos \theta,r \sin \theta,z) r dz dr d\theta$
Work Step by Step
The conversion of rectangular coordinates to spherical coordinates is given as:
$x=\rho \sin \phi \cos \theta; y=\rho \sin \phi \sin \theta;z=\rho \cos \phi$
Here, $\rho=\sqrt {x^2+y^2+z^2}$; $\phi =\cos^{-1} [\dfrac{z}{\rho}]; \theta=\cos^{-1}[\dfrac{x}{\rho \sin \phi}]$
The conversion of rectangular to cylindrical coordinate system is : $r^2=x^2+y^2 \\ \tan \theta=\dfrac{y}{x} \\z=z$
Now, $x=r \cos \theta; y=r \sin \theta, z=z$
and $\iiint f(x,y,z) dz r dr d\theta=\iiint f(r \cos \theta,r \sin \theta,z) dz r dr d\theta$
Need to plug the boundaries, then we get
$\iiint f(r \cos \theta,r \sin \theta,z) dz r dr d\theta= \int_{0}^{\pi/2}\int_{0}^{3} \int_0^{2}f(r \cos \theta,r \sin \theta,z) r dz dr d\theta$
