Oil leaks out of a tanker at a rate of $r=f(t)$ liters per minute, where $t$ is in minutes. If $f(t)=A e^{-k t}$, write a definite integral expressing the total quantity of oil which leaks out of the tanker in the first hour.
\[
\begin{array}{l}
\text { total quantity }=\int_{a}^{b} \\
\text { (where } a=
\end{array} \quad \text { and } b=
\]
Answer
The total quantity of oil which leaks out of the tanker in the first hour is given by the definite integral \(\boxed{\int_{0}^{60} A e^{-k t} dt = \frac{A}{k} - \frac{A e^{-60k}}{k}}\) if \(k \neq 0\) and \(60A\) if \(k = 0\).
Step 1 ：The total quantity of oil leaked out of the tanker in the first hour can be calculated by integrating the rate function over the time interval.
Step 2 ：In this case, the rate function is given by \(f(t)=A e^{-k t}\) and the time interval is from \(t=0\) to \(t=60\) minutes (since there are 60 minutes in an hour).
Step 3 ：Therefore, the total quantity of oil leaked out can be expressed as the definite integral \(\int_{0}^{60} A e^{-k t} dt\).
Step 4 ：The total quantity of oil which leaks out of the tanker in the first hour is given by the definite integral \(\boxed{\int_{0}^{60} A e^{-k t} dt = \frac{A}{k} - \frac{A e^{-60k}}{k}}\) if \(k \neq 0\) and \(60A\) if \(k = 0\).
