Topic 6.4 Notes – The Fundamental Theorem of Calculus and Accumulation Functions

The Fundamental Theorem of Calculus and Accumulation Functions

Take a continuous function $f$. Now define a new function:

$$F(x) = \int_a^x f(t)\,dt$$

This is an accumulation function. For each value of $x$, it gives the net signed area under $f$ from a fixed starting point $a$ to that $x$.

A definite integral usually gives a number.
If one bound is a variable, it gives a function.

Here’s the key result:

$$\frac{d}{dx} \left( \int_a^x f(t)\,dt \right) = f(x)$$

That’s the Fundamental Theorem of Calculus (Part 1).

It says:

• Integration accumulates change.
• Differentiation measures instantaneous change.
• If $f$ is continuous, these undo each other.

So every accumulation function is automatically an antiderivative of $f$.

What this looks like in action

Suppose

$$F(x) = \int_0^x e^t \, dt$$

You could evaluate the integral and get $e^x - 1$.
Or you could use the theorem immediately:

$$F'(x) = e^x$$

The derivative of the accumulation function is just the integrand with $t$ replaced by $x$.

Notice something important: the $t$ inside the integral is a dummy variable. It disappears after differentiation.

What an Accumulation Function Represents

Think of $f(x)$ as a rate.

• If $f$ is velocity → $F$ is position change.
• If $f$ is a growth rate → $F$ is total growth.
• If $f$ is flow rate → $F$ is total volume accumulated.

Here’s the visual idea. The graph shows $f(x)$ and the shaded region from a fixed starting point to a moving right endpoint $x$.

Accumulated area under $f$ from a fixed left endpoint to $x$

As $x$ moves right, the shaded area grows. The rate at which it grows at that instant is $f(x)$. That’s exactly what the theorem says: $F'(x) = f(x)$.

Connecting behavior of $F$ and $f$

Because $F'(x) = f(x)$:

• If $f(x) > 0$, then $F$ is increasing.
• If $f(x) < 0$, then $F$ is decreasing.
• Critical points of $F$ happen where $f(x) = 0$.
• Since $F''(x) = f'(x)$, concavity of $F$ depends on whether $f$ is increasing or decreasing.

This shows up a lot on FRQs. You’re often given a graph of $f$ and asked about where $F$ increases, has extrema, or changes concavity.

Representing Accumulation Functions with Integrals

You should be comfortable writing a function defined by accumulation.

Example:
Let $r(t)$ be the rate (in liters per minute) at which water flows into a tank.
Then the total water added from time 2 to time $x$ is

$$A(x) = \int_2^x r(t)\,dt$$

That’s exactly what FUN-5 is about. A definite integral can define a new function.

And as long as $r$ is continuous,

$$A'(x) = r(x)$$

On a multiple-choice question, they may give you a messy integrand. Don’t integrate it unless they ask you to. If the question is asking for the derivative, just apply the theorem.

What Has to Be True

The theorem requires:

• $f$ is continuous on an interval containing $a$.

On the AP exam, they usually state continuity. If they don’t, assume it unless the problem clearly suggests otherwise.

Common Errors I See Every Year

• Writing the derivative as $f(t)$ instead of replacing with $x$.
• Trying to actually compute the integral before differentiating.
• Forgetting that the result is about rate of accumulation, not total area.
• Mixing up definite integrals that give numbers with accumulation functions that give functions.