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
Case 1: In the spherical coordinates system, we have $x=\rho \sin \phi \cos \theta; y=\rho \sin \phi \sin \theta;z=\rho \cos \phi$
and
$\rho=\sqrt {x^2+y^2+z^2}$;
and $ \phi =\cos^{-1}(\dfrac{z}{\rho})$; $ \theta=\cos^{-1} (\dfrac{x}{\rho \sin \phi})$
Case 2: Need to apply the cylindrical coordinate system.
$r^2=x^2+y^2 ; \theta=\arctan(\dfrac{y}{x})$
and $x=r \cos \theta; y=r \sin \theta, z=z$
$\iiint f(x,y,z) dz r dr d\theta=\iiint f(r \cos \theta,r \sin \theta,z) dz r dr d\theta$
Now, use the boundaries in order to get the triple integration.
$\iiint f(r \cos \theta,r \sin \theta,z) dz r dr d\theta= \int_{0}^{\pi/2}\int_{0}^{3} \int_0^{2}f(r \cos \theta,r \sin \theta,z) r dz dr d\theta$
