A non-conducting infinite solid cylinder of radius R carries a volume charge density rho that varies with radial distance r from the axis as rho = rh…

Consider a coaxial Gaussian cylinder of radius $r$ ($r < R$) and length $L$. The charge enclosed is $q_{\text{enc}} = \int_0^r (\rho_0 r')(2\pi r' L) dr' = 2\pi L \rho_0 \int_0^r r'^2 dr' = \frac{2\pi L \rho_0 r^3}{3}$. By Gauss's law, $\oint E \cdot dA = E(2\pi r L) = \frac{q_{\text{enc}}}{\varepsilon_0} = \frac{2\pi L \rho_0 r^3}{3\varepsilon_0}$. Solving for $E$ gives $E = \frac{\rho_0 r^2}{3\varepsilon_0}$.