Topic 8.3 Notes – Using Accumulation Functions and Definite Integrals in Applied Contexts

1. What an Accumulation Function Represents

An accumulation function is defined by an integral:

$$A(x) = \int_a^x r(t)\,dt$$

Here’s what that means in plain language:

• $r(t)$ is a rate of change (like gallons per hour, meters per second, people per day).
• $A(x)$ represents the total accumulated change from time $t=a$ to $t=x$.

This connects directly to the Fundamental Theorem of Calculus:

If $Q'(t) = r(t)$, then

$$\int_a^b r(t)\,dt = Q(b) - Q(a)$$

So you can rewrite that as:

$$Q(b) = Q(a) + \int_a^b r(t)\,dt$$

That equation shows up everywhere in Unit 8.

Units Tell the Story

If

• $r(t)$ is in liters per minute,

then

• $\int r(t)\,dt$ is in liters.

The integral removes the “per.” If your units don’t simplify correctly, something’s wrong.

Net Change vs. Total Movement

The definite integral gives net change:

• Area above the x-axis → positive change
• Area below the x-axis → negative change
• Net change = positive area − negative area

If a velocity graph goes below the axis, the object is moving backward. The integral reflects that.

2. The Net Change Formula in Applied Contexts

This formula is the heart of this topic:

$$\textbf{Final amount} = \textbf{Initial amount} + \int_a^b (\textbf{rate})\,dt$$

Every applied accumulation problem fits this structure.

You need to identify:

• What quantity is changing?
• What is its rate of change?
• Over what interval?
• Is an initial value given?

Recognize These Instantly

If the problem gives a rate and asks “how much,” your brain should immediately think integral.

If it asks for the amount at time $b$, remember to include the initial value.

On AP free-response questions, forgetting that initial condition is one of the most common lost points.

3. How to Solve Accumulation Problems

Here’s the clean structure these always follow.

Step 1: Identify the rate function

Look for phrases like:

• “increases at a rate of…”
• “flowing at…”
• “growing at…”
• units that include “per”

That function is what you integrate.

Step 2: Set up the definite integral

$$\int_a^b r(t)\,dt$$

The bounds must match the time interval in the problem.

Step 3: Evaluate the integral

• Use antiderivatives (no-calculator section)
• Or use calculator integration on appropriate sections

Step 4: Add the initial value (if needed)

If the question asks for the actual amount, not just change:

$$Q(b) = Q(a) + \int_a^b r(t)\,dt$$

If it only asks for net change, stop after the integral.

4. Interpreting Accumulation Graphically

When the rate is given as a graph, the integral represents signed area.

In a picture like this, the shaded regions above the x-axis count as positive area, and the shaded regions below the x-axis count as negative area.

Signed area under a rate function

Key ideas:

• If $r(t) > 0$, the accumulated quantity increases.
• If $r(t) < 0$, the accumulated quantity decreases.
• Large positive and negative areas can cancel, giving small net change.

On FRQs, you’re often expected to:

• Break the graph into geometric shapes
• Add and subtract areas carefully
• State your answer with correct units

If the rate graph crosses the axis, always check whether you’re adding or subtracting that region.

5. Common Mistakes That Cost Points

• Treating the integral as the total amount when an initial value is given.
• Forgetting units in interpretation questions.
• Using wrong bounds.
• Mixing up displacement and total distance traveled.
• Writing the setup without explaining what the integral represents in context.

When explaining answers on FRQs, don’t just compute. Say what the integral represents in words.