Example 2.79. Determine the amount of heat $Q$ generated by the current $I=5+4 t$ in a conductor with resistance $R=40$ over time $t=10$, given that the amount of heat generated per unit time by a constant current flowing through a conductor with constant resistance is equal to the product of the square of the current and the resistance.
Solution.

$$
\begin{aligned}
& Q=\int_{t_{1}}^{t_{2}} I^{2}(t) R d t=R \int_{i_{1}}^{t_{2}} I^{2}(t) d t=40 \int_{0}^{10}(5+4 t)^{2} d t= \\
& =40 \int_{0}^{10}(5+4 t)^{2} \frac{1}{4} d(5+4 t)=\left.40 \frac{(5+4 t)^{3}}{3}\right|_{0} ^{10}=303750
\end{aligned}
$$