A very long, nonconducting cylindrical shell of inner radius \(R\) and outer radius \(2R\) has a non-uniform volume charge density given by \(\rho(r) = \beta r\) for \(R \le r \le 2R\), where \(\beta\) is a positive constant and \(r\) is the radial distance from the central axis. The regions \(r < R\) and \(r > 2R\) are empty space. The cylinder is sufficiently long that edge effects can be ignored.