Section outline

  • Photo of an iceboat in action.

    Iceboats are a common sight on the lakes of Wisconsin and Minnesota on winter weekends. Iceboats are similar to sailboats, but they are fitted with runners, or “skates,” and are designed to run over the ice, rather than on water. Iceboats can move very quickly, and many ice boating enthusiasts are drawn to the sport because of the speed. Top iceboat racers can attain speeds up to five times the wind speed. If we know how fast an iceboat is moving, we can use integration to determine how far it travels. We revisit this question later in the chapter (see "Chapter Opener: Iceboats" Example in Section 4.5: Average Value and The Net Change Theorem).

    Determining distance from velocity is just one of many applications of integration. In fact, integrals are used in a wide variety of mechanical and physical applications. In this chapter, we first introduce the theory behind integration and use integrals to calculate areas. From there, we develop the Fundamental Theorem of Calculus, which relates differentiation and integration. We then study some basic integration techniques and briefly examine some applications.

    Image Caption: Iceboating is a popular winter sport in parts of the northern United States and Europe. (credit: modification of work by Carter Brown, Flickr)

    (Content & Image Source: Chapter 5 Introduction, Calculus Volume 1, Gilbert Strang and Edwin "Jed" Herman, OpenStax, CC BY-NC-SA License)

    Upon completion of this module, you will be able to: 

    Section 4.1: The Definite Integral
    • Use geometry with rectangles to approximate area under a function
    • Use the definition of a Riemann sum to solve problems
    • Calculate area under a function using calculus notation
    • Evaluate a definite integral involving signed area
    • Use definite integrals to solve real-life problems

    Section 4.2: The Fundamental Theorem and Anti-differentiation
    • Find general antiderivatives
    • Use Net Change to calculate definite integrals
    • Use the Second Fundamental Theorem of Calculus to solve problems

    Section 4.3: Antiderivatives of Formulas
    • Use antiderivative rules to find indefinite integrals
    • Evaluate definite integrals using the Fundamental Theorem of Calculus
    • Use definite integrals to solve real-life problems

    Section 4.4: Substitution
    • Use substitution to find indefinite integrals
    • Use limit of integration substitution for definite integrals

    Section 4.5: Average Value and The Net Change Theorem
    • Find the average value of a function
    • Explain the significance of the net change theorem.
    • Use the net change theorem to solve applied problems.

    Section 4.6: Applications to Business
    • Solve consumer and producer surplus problems
    • Solve continuous income stream problems

    To achieve these objectives:
    1. Read the Module 5 Introduction (see above).
    2. Read Sections 4.1 - 4.6 of Chapter 4: The Integral in Applied Calculus (links to each Section provided below)
      • At the end of this Section there is a list of Vocabulary, a Self Check Quiz, and a set of Flashcards
    3. Watch the Videos for each Section (links provided below)
    4. Complete the MyOpenMath Homework Assignments for each Section (links provided below) - These are graded!
    5. Practice the problems on the Chapter 4 Review Exercises, checking the solutions provided (link provided below)
    6. Complete the MyOpenMath Quiz for Chapter 4 (link provided below) - This is graded!
    7. Once you complete the Quiz, upload your work in the Quiz Work Upload Assignment using the submission link below.
    8. Post in the Chapter 4 Q&A Discussion Forum - link provided below.

    Note the check boxes to the right that help you track your progress: some are automatic, and some are manual.


    Module Pressbooks Resources and Activities

    You will find the following resources and activities in this module at the Pressbooks website. Click on the links below to access or complete each item.

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