Calculate $$I = \iiint_V \frac{1}{(xyz)^2}\cdot\frac{1}{1+x^{-6}+y^{-8}+z^{-10}}\,dx\,dy\,dz$$
We know $\Omega$ is tuple $(x,y,z) \in \mathbb{R}^3$ and $x>0$ , $y>0$ , $z>0$
I don't now how to use Stokes Theorem or Gauss-Divergence Theorem. I think that i need a surface for get integration limits or $\vec{n}$.
Thank you so much
Edit:
$\Omega$ is a function limited on the entire first octant.
$\Omega$ is the function and $V$ is "Volume" closed on this function.
I understand "Calculate this Volume". I get this exercise in a Multivariable calculus test. I really don't know how to propose this problem. I think i need more information for calculate I. Maybe my teacher have been wrong?
The exactly problem was: I translated this from Spanish to English.
Calcula $$I = \iiint_V \frac{1}{(xyz)^2}\cdot\frac{1}{1+x^{-6}+y^{-8}+z^{-10}}\,dx\,dy\,dz$$ siendo $\Omega$ las ternas de números reales $(x,y,z)$ tales que $x>0$, $y>0$, $z>0$.
Edit 2:
I asked my teacher how to resolve this. He answered me that I have to make an easly variable change. Sorry for my mistake, we don't need Divergence Theorem. I will try with the following change but I think that I won't resolve this integrate.
\begin{array} & x = a \rho \sin^p{\varphi} \cos^q{\alpha} \\ y = b \rho \sin^p{\varphi} \sin^q{\alpha} \\ z = c \rho \cos^p{\varphi} \end{array}