Let F be a vector field $F = \left \langle x^3,y^3,z^3 \right \rangle $ and $\Omega$ be the solid region in $R^3$ bounded by $$x^2+y^2\ge z^2,\space x^2+y^2+z^2\le 9,\space y\ge \left | x \right | .$$ Evaluate $\iiint\limits_{\Omega}^{} \left ( x^2+y^2+z^2 \right ) dV$ and flux integral $\iint_{\partial \Omega}^{} F\cdot \overrightarrow{n} dS$ where $\partial \Omega$ is the boundary surface of $\Omega$.
Cause the "shape" of $\Omega$ is really hard to visualize, I failed to obtain the range of the integral. Is there any suggestion?
In spherical coordinates $$ x=r\sin\theta\cos\phi ; y=r\sin\theta\sin\phi ; z=r\cos\theta ; $$ The Jacobian is $r^2\sin\theta$. $x^2+y^2\ge z^2$ indicates $\sin^2\theta \ge \cos^2\theta$, therefore $\pi/4 \le \theta \le 3\pi/4$. $y\ge |x|$ indicates the 2D cone $\pi/4 \le \phi \le 3\pi/4$. $$ \int_{\Omega} (x^2+y^2+z^2) r^2\sin\theta dr d\phi d\theta $$ $$ =\int_0^3 dr \int_{\pi/4}^{3\pi/4} d\phi \int_{\pi/4}^{3\pi/4} d\theta r^2\sin\theta r^2 $$ $$ =\frac{\pi}{2}\int_0^3 dr \int_{\pi/4}^{3\pi/4} d\theta r^2\sin\theta r^2 $$ $$ =\frac{\pi}{2}\sqrt{2} \int_0^3 dr r^4 =\frac{243}{5} \sqrt{2} \pi $$