A spherical region of radius R is filled with a charge such that the charge density depends only on the radial distance r, as $\rho \left(r\right)={\rho }_0\left(\frac{5}{4}-\frac{r}{R}\right)$. The field at a distance $r(r < R)$ from the origin is given by: 

To find the field  at distance $r(r<R)$ , Let us imagine a spherical boundary of radius r (see figure). The charge in the shaded region is 

$\begin{array}{l}q={\int }_0^r {\rho }_0\left(\frac{5}{4}-\frac{x}{R}\right)\left(4\pi x^{2}dx\right)\\ =4\pi {\rho }_0{\int }_0^r \left(\frac{5x^2}{4}-\frac{x^3}{R}\right)dx=4\pi {\rho }_0\left(\frac{5r^3}{12}-\frac{r^4}{4R}\right)\end{array}$

Now, field at distance r is

$\begin{array}{l}E\left(r\right)=\frac{1}{4\pi {\epsilon }_0}\frac{q}{r^2}=\frac{1}{4\pi {\epsilon }_0}\frac{4\pi {\rho }_0}{r^2}\left(\frac{5r^3}{12}-\frac{r^4}{4R}\right)\\ =\frac{{\rho }_{0}r}{4{\epsilon }_0}\left(\frac{5}{3}-\frac{r}{R}\right)\end{array}$