7.3 Anti-Derivatives and The Fundamental Theorem of Calculus

In this lesson, you will learn to calculate antiderivatives (indefinite integrals), to apply the Fundamental Theorem of Calculus, and to use properties of integrals.

Lesson Objectives

Upon completion of the lesson 7.3, you will be able to:

• • Use derivative rules in reverse to calculate anti-derivatives.

  • Apply the Fundamental Theorem of Calculus (Part 1) to differentiate integral functions.

  • Apply the Fundamental Theorem of Calculus (Part 2) to evaluate a definite integral.

  • Recognize differentiation and integration as inverse operations.

  • Recognize that the properties of integrals apply to definite integrals as well as indefinite integrals.

Lesson Content

View all of the following instructional videos. These will help you master the objectives for this module.

• 1. YouTube video: Anti-derivatives: Definition

• 1. YouTube video: Anti-differentiation

• 1. YouTube video: Definition of Indefinite Integral

• 1. YouTube videos: Fundamental Theorem of Calculus

  2. Part 1:

• 1. Part 2:

Lesson Readings

The following required readings cover the content for this module. As you go through each reading, pay close attention to the content that will help you learn the objectives for this module.

1. Antiderivatives [Bakersfield College]

2. Indefinite Integrals [Paul's Online Math Notes]

   1. Computing Indefinite Integrals [Paul's Online Math Notes]

   2. Note: Skip the discussion of exponential, logarithmic, inverse trigonometric, and hyperbolic functions. Skip example 1f. Skip examples 2a - 2d. Skip example 4a.

   3. Definition of the Definite Integral [Paul's Online Math Notes]

   4. Note: Scroll down to the discussion of the Fundamental Theorem of Calculus Part 1. Skip the discussion of exponential, logarithmic, inverse trigonometric, and hyperbolic functions. Skip example 6a.

   5. Computing the Definite Integral [Paul's Online Math Notes]

   6. Note: Skip examples 2c and 3d.

   7. Indefinite Integrals [Nipissing University]

   8. Note: Read only the section titled Indefinite Integrals. Be sure to look at the three examples, too.

Lesson Practice Exercises/Activities

Make your way through each of the practice exercises. This is where you will take what you have learned from the lesson content and lesson readings and apply it by solving practice problems.

• 1. Antiderivatives Exercises [Bakersfield College] (with solutions)

  2. Indefinite Integrals [Paul's Online Math Notes] (with solutions)

  3. Note: Do problems #1a, 2a, 3 - 7.

  4. Computing Indefinite Integrals [Paul's Online Math Notes] (with solutions)

  5. Note: Do problems #1 - 11, 13 - 16, and #22.

  6. Definition of the Definite Integral [Paul's Online Math Notes] (with solutions)

  7. Note: Do problems #10 - 12.

  8. Computing the Definite Integral [Paul's Online Math Notes] (with solutions)

  9. Note: Do problems #1 - 5, 7 - 12, 15, 17, and 18.

  10. Indefinite Integrals [University of Houston] (with solutions)

  11. Note: Do problems #1 - 5. Since |v(t)| gives the speed, to find the distance traveled, we must find ∫ |v(t)| dt

  12. The Function [University of Houston] F(x)= ∫axf(t)dt

  13. Note: Do problems #2 - 6.

  14. The Fundamental Theorem of Integral Calculus (Part 2) [University of Houston]

  15. Note: Do problems #1 - 6.

Additional Resources

Below are additional resources that help reinforce the content for this module.

• 1. Sections 4.9: Anti-derivatives [University of Portland]

  2. Note: Skip example 1.5(i).

1. Fundamental Theorem of Calculus (Chapter 5, 5.4)

   1. The Fundamental Theorem of Calculus [University of Houston]

   2. YouTube video: Definite integrals (area under a curve) (Part III)

• 1. YouTube videos: The Fundamental Theorem of Calculus

     • Part 1

     • Part 2

     • Part 3

     • Part 4

• • Part 5