Problems with the electric field study on cylindrical symmetry surfaces using Gauss's Law

I was exploring cases of Gauss’s Law application on different surfaces in the Professor Moyses Nussenzveig book (Basic Physics Course, Vol.III) and I’m having a hard time understanding completely a specific example. Basically, there is a coaxial cable formed by a cylindrical shell of length $L$, inner radius $r1$ and outer radius $r2$. The shell has a charge $q>0 $ and the central wire has a charge $ q <0 $. I would like to find the charge densities on the inner and outer surfaces, as well as the electric field as a function of the cylinder radius.

I initially thought of finding $ vec E $ through integration, considering $vec E $//$ vec n $

$$int_Svec{mathbf{E}}cdothat{mathbf{n}}dA=E(2pi rL)+0+0=2pi rLE$$

I also know that $q_{mathrm{enc}}=lambda_{mathrm{enc}}L$, because $q_{mathrm{enc}}$ of Gauss’s law is directly proportional to L.

So, i found the field for $r <r1 =$ $frac {-lambda}{2πrepsilon0}$ and for $ r1<r<r2$ = $0$.

However, I have doubts on how I should find the field for $ r> r2 $, since there is a charge on the shell’s outer surface. Any thoughts?