Problem 5 - Applied Calculus

Question

Oil leaks out of a tanker at a rate of \(r=f(t)\) gallons per minute, where \(t\) is in minutes. Write a definite integral expressing the total quantity of oil which leaks out of the tanker in the first hour.

Step-by-Step Solution

Answer

\( \int_{0}^{60} f(t) \, dt \)

Step 1: Understand the Integral

The problem asks for the total quantity of oil leaked over the first hour. This total quantity is the accumulation of the rate of leakage over time. The rate of leakage is given by the function \( r = f(t) \). To find the total quantity of oil, we need to sum up these rates from \( t = 0 \) to \( t = 60 \) minutes. This process is accomplished using a definite integral.

Step 2: Set Up the Integral

We set up the definite integral \( \int_{0}^{60} f(t) \, dt \) to calculate the total quantity of oil that leaks out. In this integral, \( f(t) \) represents the rate of oil leakage, and the limits from 0 to 60 represent the first hour (since there are 60 minutes in an hour). This integral will provide the accumulated amount of oil leaked in that time frame.

Step 3: Express the Solution

The definite integral \( \int_{0}^{60} f(t) \, dt \) is the solution to the problem. This expression represents the total quantity of oil leaked over the first hour, given the leakage rate function \( f(t) \). The integral evaluates the continuous accumulation of the leakage rate from \( t = 0 \) to \( t = 60 \) minutes.

Key Concepts

Accumulation

When we think about accumulating something, it usually means that we are gathering together small quantities to form a larger whole. In this exercise, we are interested in the accumulation of oil leakage over time. The rate at which the oil leaks is given by the function \( r = f(t) \), where \( t \) is time in minutes.
Accumulate here means to add up the tiny amounts of oil that leak out every minute. Until, eventually, you have the total amount leaked over the entire time period. This accumulation is achieved through the process of integration.

• Accumulation refers to summing incremental changes over a set interval.
• In this specific problem, it pertains to oil quantity leaked per minute.
• Definite integrals help us compute this cumulative effect.

Hence, when you are using a definite integral, you are looking at how these small quantities add up over a given interval of time. That's why the definite integral \( \int_{0}^{60} f(t) \, dt \) is used in this context, to calculate the total oil leaking over an hour.

Rate of Change

The rate of change is a crucial concept in understanding how one quantity changes in relation to another. In our problem, it's about how the amount of oil leaked changes over each minute.
The function \( r = f(t) \) portrays this rate of change. It tells us how fast or slow the oil is leaking out at any given moment. When thinking about rates, consider how fast a car might be traveling — that's a rate (speed), and maybe it's changing (accelerating or decelerating) over time.

The rate of change:

• Describes how quantity alters over time.
• Is often denoted with a function like \( f(t) \), in this case, for oil leakage.
• Fluctuations in this rate influence the total accumulation of leaked oil.

By evaluating this rate over a specific interval, you can ascertain how much of a substance has accumulated during that time.

Integration

Integration is the mathematical process used to compute accumulation based on a given rate of change. It is the reverse operation of differentiation. While differentiation breaks functions down, integration combines them up. In our problem, we want to integrate the function \( f(t) \) over the interval from \( 0 \) to \( 60 \) minutes.
The integral \( \int_{0}^{60} f(t) \, dt \) is the tool that accumulates all the oil leaked by summing the instantaneous rates of oil leakage over each passing minute within this time frame.

• Integration acts like summing up infinitely small pieces, in this case, instantaneous leak rates.
• The definite integral has set bounds, from 0 to 60, signifying the observation period: one hour.
• The area under the curve \( f(t) \) within these bounds equals the total leaked oil.

This integral conveys the total accumulated oil over 60 minutes, making integration invaluable for situations involving continuous change.