Practice Problem Set: The Fundamental Theorem of Calculus

What the Fundamental Theorem of Calculus Does

The Fundamental Theorem of Calculus connects the two central ideas of calculus: differentiation and integration. Differentiation measures instantaneous change, while integration measures accumulated change. The theorem shows that these operations are, in an important sense, opposites.

There are two parts you need to know for AP Calculus. Part 1 tells you how to differentiate a function defined by an integral. Part 2 tells you how to evaluate a definite integral using an antiderivative.

Part 1: Differentiating an Accumulation Function

then the Fundamental Theorem of Calculus gives

The integral and derivative effectively cancel because the upper limit is simply . If the upper limit is a function such as , you must also use the chain rule.

Part 2: Evaluating a Definite Integral

If is an antiderivative of , then

This is usually written as

In practical terms, find an antiderivative, evaluate it at the upper endpoint, evaluate it at the lower endpoint, and subtract.

How to Approach AP Calculus Problems

Look for an integral. Determine whether the question is asking you to evaluate an integral or differentiate a function defined by an integral.
Check the limits of integration. A variable upper limit often signals FTC Part 1.
Watch for a function in the limit. If the upper limit is something like , the chain rule is required.
For a definite integral, find an antiderivative. Then use upper minus lower.
Interpret the result. On AP free-response questions, the integral often represents accumulated quantity, net change, displacement, or area.

Worked Examples

Example 1: Basic FTC Part 1

Let . Find .

Step 1: The upper limit of integration is simply , so FTC Part 1 applies directly.

Step 2: Differentiate by replacing the integral with its integrand evaluated at the upper limit.

Result:

Example 2: FTC Part 1 + Chain Rule

Let . Find .

Step 1: The upper limit is , so the chain rule is required.

Step 2: Evaluate the integrand at the upper limit:

Step 3: Multiply by the derivative of the upper limit, .

Result:

Example 3: FTC Part 2

Evaluate .

Step 1: Find an antiderivative:

Step 2: Apply the limits using upper minus lower:

Step 3: Simplify.

Result:

Example 4: Variable Limits on Both Ends

Let . Find .

Step 1: Use the generalized FTC rule: the derivative of an integral with variable upper and lower limits is the integrand at the upper limit times the derivative of the upper limit, minus the integrand at the lower limit times the derivative of the lower limit.

Step 2: Substitute the upper limit :

Step 3: Substitute the lower limit :

Result:

Common Mistakes

Forgetting the chain rule

If the upper limit is rather than , you must multiply by the derivative of . For example, differentiating gives , not simply .

Using lower minus upper

When evaluating a definite integral with FTC Part 2, remember upper minus lower. If , the answer is , not the reverse.

Confusing net change with total distance

A definite integral of a velocity function gives net displacement, not necessarily total distance traveled. If velocity changes sign, total distance requires integrating the absolute value or treating the intervals separately.

Adding an unnecessary constant of integration

For a definite integral, you do not need . The constant cancels when you evaluate upper minus lower.

Practice Problems

Question 1

Let . Find .

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Question 2

Let . Find .

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Question 3

Evaluate .

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Question 4

Let . Find .

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Question 5 (Interpretation)

A particle moves along the -axis with velocity . Find the particle's displacement from to .

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Question 6 (Combined FTC + Chain Rule)

Let . Find .

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AP Exam Strategy

The Fundamental Theorem of Calculus appears throughout the AP Calculus curriculum rather than as a single isolated skill. You may need it when analyzing accumulation functions, interpreting graphs, finding extrema, working with particle motion, or connecting a rate of change to a total change.

A particularly useful habit is to ask yourself what the integral represents before calculating it. If the integrand is a rate, the integral represents accumulated change. If the integrand is velocity, it represents displacement. If the integrand is a positive function, it may represent area under a curve.

For FTC Part 1 questions, look for the pattern . For FTC Part 2 questions, look for a definite integral that can be evaluated from an antiderivative.

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