For a cylinder of radius $R$ current density $J =J_0 \frac{r}{R} $, where $J_0$ is a constant and $r$ is distance from axis. Calculate total current
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$\mathrm{dI}=\mathrm{J} .2 \pi \mathrm{r} \mathrm{dr}$
$=\frac{\mathrm{J}_{0} \mathrm{r}}{\mathrm{R}} \times 2 \pi \mathrm{dr}$
$\mathrm{I}=\frac{\mathrm{J}_{0} 2 \pi}{\mathrm{R}} \int_{0}^{\mathrm{R}} \mathrm{r}^{2} \mathrm{dr}=\frac{2 \pi \mathrm{J}_{0}}{\mathrm{R}} \times \frac{\mathrm{R}^{3}}{3}=\frac{2}{3} \mathrm{J}_{0} \cdot \pi \mathrm{R}^{2}$
$=\frac{2}{3} J_{0} A$
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