Find the flux of $$\mathbf F(x, y, z) = \langle e^{z^2}, 2y + \sin(x^2z), 4z + \sqrt{x^2 + 9y^2} \rangle$$ where $S$ is the region $x^2 + y^2 \leq z \leq 8 - x^2 - y^2$.
The answer is $96\pi$.
This is the problem to find the flux of really complicating $\mathbf F(x,y,z)$. I'm struggling with this problem more than an hour. Can anyone help me to solve this problem? I'm stuck when I set double integral. It looks impossible to integrate the function $\mathbf F$.
Divergence theorem:
$$\iint F(x,y,z) \cdot\mathrm dS = \iiint \nabla\cdot F \ \mathrm dV$$
$$\nabla\cdot F = 6$$
$$\implies6\iint 8r - 2r^3 \ \mathrm dr\ \mathrm d\theta = 12\pi \left(4r^2 - \frac {2}{4}r^4\right)\Bigg|_0^2 = 96\pi$$