The cross-sectional area of one ring is given by: \[\begin{aligned} dA = 2\pi r dr\end{aligned}\] so that the current through one ring is given by: \[\begin{aligned} dI = j(r) dA = 2\pi a r^2 dr\end{a...The cross-sectional area of one ring is given by: \[\begin{aligned} dA = 2\pi r dr\end{aligned}\] so that the current through one ring is given by: \[\begin{aligned} dI = j(r) dA = 2\pi a r^2 dr\end{aligned}\] The current through the whole wire is then found by summing the currents through each ring: \[\begin{aligned} I=\int dI = \int_0^R 2\pi a r^2 dr=\frac{2}{3}\pi aR^3\end{aligned}\]
