The charge flowing through a resistance $R$ varies with time as $Q = 2t - 8t^2$. The total heat produced in the resistance is (for $0 \leq t \leq \frac{1}{8}$)
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The charge in a resistor is $Q=2 t-8 t^{2}$ and the resistance of the resistor is $R$
The current is given as,
$I=\frac{d Q}{d t}$
$=2-16 t$
The heat produced in the resistor is given as,
$H=\int I^{2} R d t$
$=\int_{0}^{\frac{1}{8}}(2-16 t)^{2} R d t$
$=\left[4 t+\frac{256 t^{3}}{3}-32 t^{2}\right]_{0}^{\frac{1}{8}} R$
$=\frac{R}{6} \mathrm{J}$
Thus, the total heat produced is $\frac{R}{6} \mathrm{J}$
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