The charge flowing through a resistance R varies with time t as Q =...

given

$Q = at - b{t}^{2}$

$i = \dfrac{dq}{dt} = a - 2bt$

$i = 0$

at $\text{t} = \dfrac{\text{a}}{\text{2b}}$

$∴ dH = {i}^{2}Rdt$

$= {\left(a - 2bt\right)}^{2}Rdt$

$H = \mathop{∫}_{0}^{\dfrac{a}{2b}}{\left(a - 2bt\right)}^{2}Rdt$

{\left= \dfrac{{\left(a - 2bt\right)}^{3}R}{3\left(- 2b\right)}\right|}_{0}^{\dfrac{a}{2b}}

$= \dfrac{1}{-6 b}\left[{\left(a - 2b\dfrac{a}{2b}\right)}^{3} - {\left(a\right)}^{3}\right]R$

$= - \dfrac{1}{6b}\left[{\left(0\right)}^{3} - {a}^{3}\right]R$

$\text{H} = \dfrac{{\text{a}}^{\text{3}} \text{R}}{6\text{b}}$