Solving Gauss's Law for Constant Electric Field: q, a, b

[Minakami]

Homework Statement

The spherical region a < r < b carries a charge per unit volume of \rho = A/r, where A is a constant. At the center (r = 0) of the enclosed cavity is a point charge q. What should be the value of A so that the electric field in the region a< r < b has constant magnitude?

The answer says A = q/(2pi a²)

2. The attempt at a solution
I drew a sphere with radius r as a Gaussian surface.
My work is in the attachment.

I got stuck from that point. I need help :!)

 

[borgwal]

You go wrong in the second line, the right-hand side: the volume element is NOT 2\pi r dr, but...

Otherwise, you got the right idea.