$\displaystyle\iiint_R (x^2+y^2+z^2)^{-2}\,dx\,dy\,dz$ where $R$ is in the region in the first octant outside the sphere $x^2+y^2+z^2 = 1$;
Hi guys, I don't quite get which region is this, is it that I find the volume of this sphere and minus by the volume of the first octant? Thanks in advance.
No. This has nothing to do with computing volumes. The idea is to compute the integral of $(x^2+y^2+z^2)^{-2}$ in the region of the first octant outside the unit sphre. Turning this into spherical coordinates, this becomes$$\int_1^\infty\int_0^{\frac\pi2}\int_0^{\frac\pi2}\frac{\rho^2\sin\varphi}{\rho^4}\,\mathrm d\theta\,\mathrm d\varphi\,\mathrm d\rho.$$Can you take it from here?