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Fundamental Theorem of Calculus Practice Problems

For definite integrals, find an antiderivative and subtract: upper limit minus lower limit. For derivatives of integrals, use Part 1 and the chain rule. Type functions with parentheses, as in sin(x^3) or sqrt(x), and exact numbers like e - 1 or decimals rounded to three places.

  1. 1.

    Evaluate .

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    .

  2. 2.

    Evaluate .

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    .

  3. 3.

    Evaluate .

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    .

  4. 4.

    Evaluate .

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    .

  5. 5.

    Evaluate .

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    .

  6. 6.

    Evaluate .

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    , so .

  7. 7.

    Find .

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    By Part 1 of the fundamental theorem, replace with in the integrand: .

  8. 8.

    Find .

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    The upper limit is , so by the chain rule the derivative is .

  9. 9.

    Find .

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    Reverse the limits: . By Part 1 the derivative is .

  10. 10.

    Let . Find .

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    By Part 1, , so .

  11. 11.

    A particle moves with velocity m/s. Find its displacement from to , in meters.

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    By the net change theorem, displacement m. The particle ends where it started.

  12. 12.

    For the same particle, , find the total distance traveled from to , in meters.

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    is negative on and positive on . With : and . The distance is m.
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Fundamental Theorem of Calculus Practice Problems

These problems follow the fundamental theorem of calculus lesson. Problems 1–6 use Part 2 to evaluate definite integrals, 7–10 use Part 1 to differentiate integrals, and 11–12 apply the net change theorem.

The method

  1. Part 2: find an antiderivative , then compute .
  2. Part 1: . If is the lower limit, reverse the limits and change the sign.
  3. Net change: the integral of a rate is the net change; for distance, integrate .

Common mistakes to watch for

  • Subtracting in the wrong order. It’s upper limit minus lower limit.
  • Forgetting the chain rule factor when the upper limit is a function of .
  • Using displacement for distance when the velocity changes sign.

Frequently Asked Questions

Do I need + C in these answers?+

No. Definite integrals are numbers, and the derivative problems ask for a function of x with no constant.

How do I type the derivative answers?+

Use ^ for powers and parentheses for function inputs: 3x^2 sin(x^3), sqrt(x), -sqrt(x). Any equivalent form is accepted.

More practice problems

Definite Integral Practice Problems Antiderivatives Practice Problems Chain Rule Practice Problems