want to evaluate $$\int\frac{1}{\sqrt{x^2+y^2+z^2}}dxdydz$$ over entire $\mathbb{R}^3$ except $(0,0,0)$.
I did this using polar coordinate and got $$\lim\limits_{a\rightarrow0}\int\limits_a^\infty\int\limits_0^{\pi}\int\limits_0^{2\pi}r\sin\phi d\theta d\phi dr$$ but I think this diverges.
where am I wrong? help me please
What you seem to be attempting is to set $r \rightarrow \infty$ and to calculate the volume of a sphere excluding the origin. This volume is indeed $\Bbb{R}^3-{\text{{(0,0,0)}}}$.
A more compact way of writing what you're attempting to find needs set notation, for instance.
Let $I = [0,\infty),r\in I$ and $A_r=\Bbb{\text{{(x,y,z)$\in$ $\Bbb{R}^3 |x^2+y^2+z^2>r^2$}}}$
The way to imagine this is that we're essentially creating every possible sphere and the volume is all the region outside those spheres. By integrating to infinity but excluding the origin you're taking the union of all possible spheres in $\Bbb{R}^3$ excluding the origin.
$\bigcup\limits_{r\in I} A_{r}$=$\Bbb{R}^3-{\text{{(0,0,0)}}}$
By taking the union of these sets we get the answer we expected. I would attempt to draw you a diagram for this problem but I hope this suffices.