Problem 7 - 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

The total oil leaked is expressed by the integral \( \int_{0}^{60} f(t) \, dt \).

Step 1: Understanding the Problem

The problem gives a rate function, \( r = f(t) \), which represents the rate at which oil leaks from the tanker in gallons per minute as a function of time \( t \) (minutes). We need to find the total quantity of oil that leaks out in the first hour.

Step 2: Identifying the Interval

Since time \( t \) is given in minutes and the problem asks about the first hour, our time interval will be from 0 minutes to 60 minutes \( (t = 0 \text{ to } t = 60) \).

Step 3: Setting Up the Definite Integral

The total quantity of oil leaked over a given time interval can be found by integrating the rate function \( f(t) \) over the interval. Here, the definite integral is set up as: \[ \int_{0}^{60} f(t) \, dt \]

Step 4: Interpreting the Integral

The definite integral \( \int_{0}^{60} f(t) \, dt \) represents the accumulation of the rate \( f(t) \) over the time interval from 0 to 60 minutes. This gives the total quantity of oil leaked in gallons during that hour.

Key Concepts

Rate of Change

In simple terms, the rate of change tells us how quickly something happens. In our exercise, the rate of change is expressed as the function \(r = f(t)\). This function describes how fast the oil is leaking from the tanker with respect to time, measured in gallons per minute.

You can think of the rate of change as a speedometer for the leakage. If \(f(t)\) is increasing, the speed of the leak is getting faster. If it is constant, the speed of the leak is steady.

Understanding the rate of change is crucial because it helps us determine how much oil is spilled over a specific period. You'll often use the rate of change to find out the differences in quantity over time, like the total gallons of oil leaked after a certain number of minutes.

• Defines how changes occur over time
• Expressed in this case as \(f(t)\), the leak rate
• Measured in gallons per minute

Reviewing the rate of change helps us set up integrals that can predict or assess total accumulated changes over time.

Accumulation Function

The accumulation function helps us understand how much has changed up to a certain point. In context, it allows us to find out the total amount of oil that has leaked over a period of time.

When dealing with a continuous rate function like \(f(t)\), the accumulation is found using integration. Essentially, integration "collects up" the total effect of the change over time.

In our exercise, the definite integral from 0 to 60 represents the accumulation of the leak rate over those 60 minutes. By evaluating this integral, we calculate the total gallons of oil that have leaked from the tanker.

• Integrates the rate of change over time
• Represents total quantity from a start to an end point
• Key to determining overall change

Balancing the idea of rate and accumulation lets us translate rates into meaningful total quantities.

Integration

Integration is like adding up small pieces to make a whole. In our exercise, it tells us the total quantity of oil leaked over an hour by summing up all the tiny rates of leaks every minute.

The definite integral \(\int_{0}^{60} f(t) \, dt\) from the problem is a tool we use to compute this total change. Think of integration as a way to "add" the continuous amounts described by \(f(t)\) over the interval from \(t = 0\) to \(t = 60\).

Integration delivers powerful insights by translating dynamic and continuously changing rates into comprehensible totals.

• Breaks down a continuous process into manageable parts
• Turns rates into overall totals
• Essential for calculating sums in calculus

This is why it's so valuable not just in mathematics, but in applications where cumulative totals from varying rates are needed.