Answer
$\dfrac{3 \pi}{2}$
Work Step by Step
The Divergence Theorem states that $\iint_S \overrightarrow{F}\cdot d\overrightarrow{S}=\iiint_Ediv \overrightarrow{F}dV $
Here, $S$ is a closed surface and $E$ is the region inside that surface.
$div F=\dfrac{\partial a}{\partial x}+\dfrac{\partial b}{\partial y}+\dfrac{\partial c}{\partial z}=\dfrac{\partial (z \tan^{-1} y^2)}{\partial x}+\dfrac{\partial (z^3(\ln (x^2+1))}{\partial y}+\dfrac{\partial (z)}{\partial z}=0+0+1=1$
$\iiint_Ediv \overrightarrow{F}dV=\int_{0}^{2 \pi}\int_0^{1} \int_{1}^{2-r^2} r dz dr d\theta=(2 \pi) \int_{0}^{1} [zr] dr$
$=(2 \pi) \int_0^{1} r-r^3 dr$
$=\dfrac{\pi}{2}$
Now, flux through the disk $=-\iint_{D} z dA=-\pi(1)^2=-\pi$
Flux through paraboloid = Total Flux -Flux Through Disk
$=\dfrac{\pi}{2}-(-\pi)=\dfrac{3 \pi}{2}$
