Let there be a spherically symmetric charge distribution with charge density varying as $$\rho \left( r \right) = {\rho _0}\left( {\frac{5}{4} - \frac{r}{R}} \right)$$     upto $$r = R,$$  and $$\rho \left( r \right) = 0$$   for $$r > R,$$  where $$r$$ is the distance from the origin. The electric field at a distance $$r\left( {r < R} \right)$$   from the origin is given by

Solution :
Let us consider a spherical shell of radius $$x$$ and thickness $$dx.$$
Charge on this shell
$$dq = \rho .4\pi {x^2}dx = {\rho _0}\left( {\frac{5}{4} - \frac{x}{R}} \right).4\pi {x^2}dx$$
$$\therefore $$ Total charge in the spherical region from centre to $$r\left( {r < R} \right)$$   is
$$q = \int {dq = 4\pi {\rho _0}\int\limits_0^r {\left( {\frac{5}{4} - \frac{x}{R}} \right){x^2}dx} } $$

$$ = 4\pi {\rho _0}\left[ {\frac{5}{4}.\frac{{{r^3}}}{3} - \frac{1}{R}.\frac{{{r^4}}}{4}} \right] = \pi {\rho _0}{r^3}\left( {\frac{5}{3} - \frac{r}{R}} \right)$$
$$\therefore $$ Electric field at $$r,E = \frac{1}{{4\pi { \in _0}}}.\frac{q}{{{r^2}}}$$
$$ = \frac{1}{{4\pi { \in _0}}}.\frac{{\pi {\rho _0}{r^3}}}{{{r^2}}}\left( {\frac{5}{3} - \frac{r}{R}} \right) = \frac{{{\rho _0}r}}{{4{ \in _0}}}\left( {\frac{5}{3} - \frac{r}{R}} \right)$$