we know that

$J=\frac{\text {current}}{\text {area}}$

$\Rightarrow$ current $=J$ Area

current $=J_{1} A_{1}+J_{2} A_{2}$

$=J_{0}\left(\frac{x}{R}-1\right) \times(2 \pi x d x)+J_{0} \frac{x}{R}(2 \pi x d x)$

$=\int_{0}^{R / 2} J_{0}\left(\frac{x}{R}-1\right) \times(2 \pi x d x)+\int_{R / 2}^{R} J_{0} \times 2 \pi \frac{x^{2}}{R} d x$

$=J_{0} \times 2 \pi \int_{0}^{R / 2}\left(\frac{x^{2}}{R}-x\right) d x+\frac{J_{0} \times 2 \pi}{R} \int_{R / 2}^{R} x^{2} d x$

$\Rightarrow i=J_{0} \times 2 \pi\left[\frac{x^{3}}{3 R}-\frac{x^{2}}{2}\right]_{0}^{R / 2}+\frac{J_{0} \times 2 \pi}{R}\left[\frac{x^{3}}{3}\right]_{R / 2}^{R}$

$\Rightarrow i=J_{0} \times 2 \pi\left[\frac{R^{2}}{24}-\frac{R^{2}}{8}\right]+\frac{J_{0} \times 2 \pi}{R}\left[\frac{R^{3}}{3}-\frac{R^{3}}{24}\right]$

$=J_{0} \times 2 \pi\left(\frac{-2 R^{2}}{24}\right)+\frac{J_{0} \times 2 \pi}{R}\left[\frac{7 R^{3}}{24}\right]$

$i=\frac{5}{12} \pi J_{0} R^{2}$