Using Accumulation to Solve Rate-In/Rate-Out Problems

The Basic Idea: What Goes In Minus What Comes Out

Many AP Calculus problems describe a quantity that changes because material, water, people, money, or some other quantity is entering and leaving a system. These are rate-in/rate-out problems.

The key idea is simple: the quantity in the system increases when more comes in than goes out, and decreases when more goes out than comes in.

If is the rate at which material enters and is the rate at which material leaves, then the net rate of change is:

Once you know the net rate, integration tells you how much the quantity changes over a time interval.

The Accumulation Formula

Suppose a tank contains units of water at time . If water enters at rate and leaves at rate , then the amount of water at time is:

The integral represents the net accumulated change between and . You start with the original amount and add the total net amount that entered the system.

This is one of the most important interpretations of a definite integral in AP Calculus:

So if the rate is measured in gallons per minute and time is measured in minutes, the integral has units of gallons.

A Step-by-Step Method

1. Identify the initial amount. Find the quantity in the system at the beginning of the interval.
2. Identify the rate in and rate out. Pay close attention to the units and what each function represents.
3. Build the net rate. Use .
4. Integrate the net rate over the appropriate time interval.
5. Add the accumulated change to the initial amount.
6. Check the units. The final answer should be in units of the quantity being measured, not units of rate.

Worked Examples

The Most Important Interpretation: Net Accumulation

It is easy to think of an integral as simply "something you calculate." On the AP exam, the interpretation is often more important than the calculation.

If is the net rate of change, then:

This tells you that the definite integral gives the net change in the quantity. It does not necessarily give the final amount by itself.

To obtain the final amount, use:

Rate In vs. Rate Out: Keep the Signs Straight

One of the easiest ways to make a mistake is to use the wrong sign for the outflow.

If represents the rate entering and represents the rate leaving, then the net rate is:

You do not need to make the outflow function negative yourself if it is already described as a positive rate of leaving. The subtraction accounts for the fact that material is leaving the system.

When Is the Quantity Increasing or Decreasing?

The same rate-in/rate-out setup can answer questions about when the amount is increasing or decreasing.

Let be the amount in the system. Then:

Therefore:

• If , the amount is increasing.
• If , the amount is decreasing.
• If , the amount is momentarily constant.

This is especially useful for questions asking when a tank has its maximum or minimum amount. Look for where the net rate changes sign.

A Useful AP Calculus Shortcut

When you see a problem involving material entering and leaving a system, immediately write:

Then ask three questions:

1. What is the initial amount?
2. What is the net rate?
3. What time interval should I accumulate over?

Once those three pieces are identified, most rate-in/rate-out accumulation problems become straightforward.

Practice Problems