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Chapter 2: Problem 42

If oil leaks from a tank at a rate of \(\mathrm{r}(\mathrm{t})\) litres per minute at time \(t\), what does \(\int_{0}^{120} r(t) d t\) represent?

Short Answer

Expert verified
Answer: The expression \(\int_{0}^{120} r(t) dt\) represents the total amount of oil leaked from the tank over a period of 120 minutes (2 hours), given the rate of leakage \(r(t)\) litres per minute.

Step by step solution

01

Understanding the integral

Recall that integrating a function \(f(t)\) with respect to \(t\) between the limits \(a\) and \(b\) represents finding the signed area under the curve of \(f(t)\) between the two points \(t=a\) and \(t=b\). In this scenario, given the rate of oil leakage \(r(t)\), the integral \(\int_{0}^{120} r(t) dt\) represents the area under the curve of the leakage rate between the time \(t=0\) and \(t=120\) minutes.
02

Interpretation of the integral in the context of the given problem

Since the given function \(r(t)\) represents the rate of oil leakage per minute, by finding the area under this function's curve, we determine the total amount of oil leakage. Specifically, the expression \(\int_{0}^{120} r(t) dt\) gives us the total amount of oil that leaks from the tank within 120 minutes (2 hours).
03

Concluding remarks

In summary, the integral \(\int_{0}^{120} r(t) dt\) represents the total amount of oil leaked from the tank over a period of 120 minutes (2 hours), given the rate of leakage \(r(t)\) litres per minute.

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