$\mathrm{I}=\frac{\mathrm{dQ}}{\mathrm{dt}}=\mathrm{a}-2 \mathrm{bt}$

$I=0$ for $t=t_{0}=(a / 2 b)$

Current flows from $t=0$ to $t=t_{0}$

Heat produced $ = \int_0^{{t_0}} {{{\rm{I}}^2}} {\rm{R\,dt}}$

$ = \int_0^{{t_0}} {{{(a - 2bt)}^2}} R\,dt$

$\int_0^{{t_0}} {\left( {{a^2}R - 4abRt + 4{b^2}R{t^2}} \right)} dt$

Solving above equation and putting $t_{0}=(a / 2 b)$

we get Heat produced $=\left(\frac{a^{3} R}{6 b}\right)$