• Unit 5: The Integral

    While previous units dealt with differential calculus, this unit starts the study of integral calculus. As you may recall, differential calculus began with developing the intuition behind the notion of a tangent line. Integral calculus begins with understanding the intuition behind the idea of an area. We will be able to extend the notion of the area and apply these more general areas to various problems. This will allow us to unify differential and integral calculus through the Fundamental Theorem of calculus. Historically, this theorem marked the beginning of modern mathematics and is extremely important in all applications.

    Completing this unit should take you approximately 10 hours.

    • Upon successful completion of this unit, you should be able to:

      • use sigma notation to represent sums;
      • approximate the areas of regions;
      • approximate areas by Riemann sums;
      • translate an area under a curve into a definite integral;
      • evaluate definite integrals geometrically using graphs of functions;
      • determine whether a given function is integrable;
      • use antiderivatives to evaluate a function;
      • find antiderivatives by changing the variable and using tables;
      • use the Fundamental Theorem of calculus to evaluate definite integrals;
      • differentiate integrals;
      • solve applied problems that involve generalized area such as distance or work;
      • find an area between two curves; and
      • find the average (mean) value of a function.

    • 5.1: Introduction to Integration

      In this section, you will learn about the concept of Integration. Integral calculus applies to a wide range of applied problems, such as area.

    • 5.2: Sigma Notation and Riemann Sums

      Earlier, you learned about the concept of area. In this section, you will learn how to use notation to add a large set of values.

    • 5.3: The Definite Integral

      In this section, you will explore the Riemann Sums further, which will lead us to the concept of the definite integral. You will then learn about their applications.

    • 5.4: Properties of the Definite Integral

      In this section, you will continue to explore the definite integral by learning about the properties of integrals that sometimes define other functions.

      • Properties of the Definite Integral Book

        Read this section to learn about properties of definite integrals and how functions can be defined using definite integrals. Work through practice problems 1-5.

      • Integrals and the Graph of a Function Page

        Watch this video on definite integral and a function's graph.

      • Practice Problems Book

        Work through the odd-numbered problems 1-51. Once you have completed the problem set, check your answers for the odd-numbered questions against here.

    • 5.5: Areas, Integrals, and Antiderivatives

      In this section, you will explore the connection between areas, integrals, and antiderivatives, and how those connections can be used.

      • Areas, Integrals, and Antiderivatives Book

        Read this section to learn about the relationship among areas, integrals, and antiderivatives. Work through practice problems 1-5.

      • Integration and Antiderivatives Page

        Watch this video on Power Rule of Integration, Antiderivative of Polynomial Functions, Integrating Square Root Functions, and Antiderivatives of Rational Functions.

      • Practice Problems Book

        Work through the odd-numbered problems 1-25. Once you have completed the problem set, check your answers.

    • 5.6: The Fundamental Theorem of Calculus

      The Fundamental Theorem of Calculus is one that you will use very often in calculating integrals.

    • 5.7: Finding Antiderivatives

      In this section, you will apply previous general knowledge of antiderivatives to find antiderivatives of complicated functions.

      • Finding Antiderivatives Book

        Read this section to see how you can (sometimes) find an antiderivative. In particular, we will discuss the change of variable technique. Change of variable, also called substitution or u-substitution (for the most commonly-used variable), is a powerful technique that you will use time and again in integration. It allows you to simplify a complicated function to show how basic rules of integration apply to the function. Work through practice problems 1-4.

      • Integrating with U-Substitution Page

        Watch these videos on change of variable, also called substitution or U-substitution.

      • Practice Problems Book

        Work through the odd-numbered problems 1-69. Once you have completed the problem set, check your answers.

    • 5.8: First Application of Definite Integral

      Using your knowledge of definite integrals, we will now take a look at how to solve applied problems.

      • First Application of Definite Integral Book

        Read this section to see how some applied problems can be reformulated as integration problems. Work through practice problems 1-4.

      • Practice Problems Book

        Work through the odd-numbered problems 1-41. Once you have completed the problem set, check your answers.

    • 5.9: Using Tables to Find Antiderivatives

      In this section, you will learn how to use the tables of integrals. You will find the table of integrals useful in learning calculus. Think about the table of integrals as a guide to help you solve problems.

      • Using Tables to Find Antiderivatives Book

        Read this section to learn how to use tables to find antiderivatives. See the Calculus Reference Facts for the table of integrals. Work through practice problems 1-5.

      • Practice Problems Book

        Work through the odd-numbered problems 1-55. Once you have completed the problem set, check your answers.

    • Unit 5 Assessment

      • Unit 5 Assessment Quiz
        Receive a grade

        Take this assessment to see how well you understood these concepts.

        • This assessment does not count towards your grade. It is just for practice!
        • You will see the correct answers when you submit your answers. Use this to help you study for the final exam!
        • You can take this assessment as many times as you want, whenever you want.