A spherically symmetric charge distribution is considered with...

According to Gauss's law, the electric flux through a closed surface area is equal to the charge inside the surface divided by ${ε}_{0}$.

$⇒ ∮ \overset{\to}{E \cdot}d\overset{\to}{s} = \dfrac{{Q}_{in}}{{ε}_{0}}$

$⇒ E4\pi{r}^{2} = \dfrac{{∫}_{0}^{r}{ρ}_{0}\left(\dfrac{3}{4} - \dfrac{r}{R}\right)4\pi{r}^{2}dr}{{ε}_{0}}$

$⇒ E4\pi{r}^{2} = \dfrac{{ρ}_{o}4\pi}{{ε}_{o}}\left(\dfrac{3}{4}\dfrac{{r}^{3}}{3} - \dfrac{{r}^{4}}{4R}\right)$

$⇒ E = \dfrac{{ρ}_{o}r}{4{ε}_{o}}\left(1 - \dfrac{r}{R}\right)$