The Fundamental Theorem of Calculus, Part 2

What the Fundamental Theorem of Calculus, Part 2 Does

The Fundamental Theorem of Calculus, Part 2 gives you a practical way to evaluate a definite integral. Instead of thinking about an integral as the limit of infinitely many rectangles, you can find an antiderivative and evaluate it at the two endpoints.

If is an antiderivative of , then:

This is one of the most important formulas in AP Calculus. It connects the two major ideas of calculus: derivatives and integrals. Differentiation finds rates of change; integration accumulates those rates of change.

The Formula: "Upper Minus Lower"

The notation can look more intimidating than it really is. The expression

means exactly:

In other words, evaluate the antiderivative at the upper limit, evaluate it at the lower limit, and subtract.

A Step-by-Step Method

1. Find an antiderivative. Use your integration rules to find a function whose derivative is the integrand.
2. Write the antiderivative at the upper endpoint. Substitute the upper limit into the antiderivative.
3. Write the antiderivative at the lower endpoint. Substitute the lower limit into the antiderivative.
4. Subtract: upper minus lower. This is the most important sign rule to remember.
5. Simplify. Your final answer is the value of the definite integral.

Worked Examples

Why There Is No "+ C" in a Definite Integral

When you find an indefinite integral, you write because there are infinitely many antiderivatives that differ by a constant.

For a definite integral, however, the constant cancels:

So you normally do not write when evaluating a definite integral with the Fundamental Theorem of Calculus.

Understanding the Geometric Meaning

The value of a definite integral represents net signed area between the graph of the function and the -axis over the interval.

If the function is above the axis, the integral contributes positive area. If the function is below the axis, it contributes negative area.

This is why the result of a definite integral can be negative even though ordinary geometric area cannot be negative. The integral measures signed accumulation.

When the Function Crosses the x-Axis

Suppose a function changes sign inside the interval. The Fundamental Theorem still works exactly the same way. You can evaluate the entire integral using one antiderivative.

But be careful: if the question asks for total area rather than net area, you may need to split the interval at the points where the function crosses the -axis.

Using the Fundamental Theorem in AP Problems

On the AP exam, you will often see the theorem used in situations where the antiderivative is straightforward. But the real skill being tested is often whether you understand what the integral represents.

For example, if represents velocity, then:

The integral gives the object's change in position. If the velocity becomes negative, the object is moving in the opposite direction, so that part of the integral subtracts from the total displacement.

Common Mistakes

Practice Problems