Answer
$\dfrac{1}{8}$
Work Step by Step
Region of integration:
$R=${$ (r,\theta, z) | r^2 \lt z \leq 1, 0 \leq r \leq 1, 0 \leq \theta \leq 2 \pi$}
Consider $I= \iint_{R} |xyz| \ dV$
or, $=\int^{0}_{2 \pi} \int_{0}^{1} \int_{r^2}^1 |(r \cos \theta) (r \sin \theta) \cdot z| \times ( r dz dr d \theta) $
or, $=\int_0^{2 \pi} \int_0^{1} \int_{r^2}^1 r^3 z \ |\sin \theta \cos \theta| \times dz dr d \theta$
We will use a calculator to determine the triple integral.
$\int_0^{2 \pi} \int_0^{1} \int_{r^2}^1 r^3 z \ |\sin \theta \cos \theta| dz dr d \theta=\dfrac{1}{8}$
