Answer
$\dfrac{1}{8}$
Work Step by Step
The region of integration in cylindrical coordinates can be expressed as:
$R=${$ (r,\theta, z) | r^2 \lt z \leq 1, 0 \leq r \leq 1, 0 \leq \theta \leq 2 \pi$}
The function we want to integrate can be integrated in triple cylindrical coordinates as:
$ \iint_{R} |xyz| \ dV=\int^{0}_{2 \pi} \int_{0}^{1} \int_{r^2}^1 |(r \cos \theta) (r \sin \theta) \cdot z| ( r dz dr d \theta) \\=\int_0^{2 \pi} \int_0^{1} \int_{r^2}^1 r^3 z |\sin \theta \cos \theta| dz dr d \theta$
We need to use a calculator to compute the triple integral.
$\int_0^{2 \pi} \int_0^{1} \int_{r^2}^1 r^3 z |\sin \theta \cos \theta| dz dr d \theta=\dfrac{1}{8}$
