We are going to use Gauss's Law: http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elecyl.html
If color(red) (r <= R)
V_r - V_R = int_r^R E_(r) dr
(V_r - V_R ) is the Electrical potential http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elepe.html
E_(r) = (Lambdar)/ (2pivarepsilon_0R^2)
int_r^R (Lambdar)/ (2pivarepsilon_0R^2) dr
Lambda/ (2pivarepsilon_0R^2) int_r^Rrdr
Lambda/ (2pivarepsilon_0R^2) [r^2/2]_r^R
Lambda/ (2pivarepsilon_0R^2) [r^2/2]_r^R
Lambda/ (2pivarepsilon_0R^2) [R^2-r^2]
V_r - V_R = Lambda/ (2pivarepsilon_0) [1-r^2/R^2]
or
If color(red) (r >= R)
E_(r) = Lambda/ (2pivarepsilon_0r)
V_r - V_R = int_r^R E_(r) dr
int_r^R Lambda/ (2pivarepsilon_0r)
Lambda/ (2pivarepsilon_0R^2)int_r^R (1/r)
Lambda/ (2pivarepsilon_0R^2) [ln(r)_r^R]
V_r - V_R = Lambda/ (2pivarepsilon_0R^2) [ln(R/r)]