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
Conversion of rectangular to spherical coordinates is as follows:
$x=\rho \sin \phi \cos \theta; y=\rho \sin \phi \sin \theta;z=\rho \cos \phi$
and
$\rho=\sqrt {x^2+y^2+z^2}$;
$\cos \phi =\dfrac{z}{\rho}$; $\cos \theta=\dfrac{x}{\rho \sin \phi}$
Conversion of the rectangular to cylindrical coordinate system is as follows:
$r^2=x^2+y^2 \\ \tan \theta=\dfrac{y}{x} \\z=z$
Here, $x=r \cos \theta; y=r \sin \theta, z=z$
Here, we have $\iiint f(x,y,z) dz r dr d\theta=\iiint f(r \cos \theta,r \sin \theta,z) dz r dr d\theta$
Plug in the boundaries:
$\iiint f(r \cos \theta,r \sin \theta,z) dz r dr d\theta= \int_{0}^{\pi/2}\int_{0}^{3} \int_0^{2}f(r \cos \theta,r \sin \theta,z) r dz dr d\theta$
