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# MA101: Single-Variable Calculus I

Unit 7: Integration   In the last unit of this course, you will learn about “integral calculus,” a subfield of calculus that studies the area formed under the curve of a function.  Although its relationship with the derivative is not necessarily intuitive, integral calculus is closely linked to the derivative, which you will revisit in this unit.

Unit 7 Time Advisory
This unit should take you approximately 15.25 hours to complete.

☐    Subunit 7.1: 3 hours

☐    Subunit 7.2: 5.5 hours ☐    Reading: 1.5 hours

☐    Lecture: 2 hours

☐    Assignment: 2 hours

☐    Subunit 7.3: 2.5 hours

☐    Subunit 7.4: 4.25 hours ☐    Reading: 1.5 hours

☐    Lecture: 0.75 hours

☐    Assignment: 2 hours

Unit7 Learning Outcomes
Upon successful completion of this unit, the student will be able to:
- Define antiderivatives and the indefinite integral. - State the properties of the indefinite integral. - Relate the definite integral to the initial value problem and the area problem. - Set up and calculate a Riemann sum. - State the Fundamental Theorem of Calculus and use it to calculate definite integrals. - State and apply basic properties of the definite integral. - Use substitution to compute definite integrals.

7.1 Motivation   - Reading: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Section 7.1: Two Examples” Link: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Section 7.1: Two Examples” (PDF)

Instructions: Please click on the link above and read Section 7.1 (pages 145-149) in its entirety.  This reading introduces the integral through two examples.  The first example addresses the question of how to determine the distance traveled based only on information about velocity.  The second example addresses the question of how to determine the area under the graph of a function.  Surprisingly, these two questions are closely related to each other and to the derivative.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  This text was originally written by Professor David Guichard.  It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills.  You can access the original version here.

• Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 18: Definite Integrals” Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 18: Definite Integrals” (YouTube)

Instructions: Please click on the link above and watch the entire video (47:14).  Lecture notes are available here.

Viewing this lecture and taking notes should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  It is attributed to David Jerison and MIT's OpenCourseWare.  It may be viewed in its original form here.

• Assignment: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Exercises 7.1: Problems 1-8” Link: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Exercises 7.1: Problems 1-8” (PDF)

Instructions: Please click on the link above and work through problems 1-8 for Exercises 7.1.  When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  This text was originally written by Professor David Guichard.  It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills.  You can access the original version here.

7.2 The Fundamental Theorem of Calculus   Note: The Fundamental Theorem of Calculus is the apex of our course.  It explains the relationship between the derivative and the integral, tying the two major facets of this course together.  In the previous section, you learned the definition of the definite integral as a limit of a Riemann Sum.  The computations were long and involved.  In this subunit, you will learn about the Fundamental Theorem of Calculus, which makes the computation of definite integrals significantly easier.

• Reading: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Section 7.2: The Fundamental Theorem of Calculus” Link: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Section 7.2: The Fundamental Theorem of Calculus” (PDF)

Instructions: Please click on the link above and read Section 7.2 (pages 149-155) in its entirety.  Pay close attention to the treatment of Riemann sums, which lead to the definite integral.  The Fundamental Theorem of Calculus explicitly describes the relationship between integrals and derivatives.

This reading should take you approximately one hour and 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  This text was originally written by Professor David Guichard.  It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills.  You can access the original version here.

• Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 19: The First Fundamental Theorem” Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 19: The First Fundamental Theorem” (YouTube)

Instructions: Watch this video lecture. Lecture notes are available here.

Watching this lecture and taking notes should take you approximately one hour.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License. It is attributed to David Jerison and MIT's OpenCourseWare. It may be viewed in its original form here.

• Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 20: The Second Fundamental Theorem” Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 20: The Second Fundamental Theorem” (YouTube)

Instructions: Please click on the link above and watch the entire video (49:30).  Lecture notes are available here.

Viewing this lecture and taking notes should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  It is attributed to David Jerison and MIT's OpenCourseWare.  It may be viewed in its original form here.

• Assignment: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Exercises 7.2: Problems 7-22” Link: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Exercises 7.2: Problems 7-22” (PDF)

Instructions: Please click on the link above and work through problems 7-22 for Exercises 7.2.  When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately two hours to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  This text was originally written by Professor David Guichard.  It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills.  You can access the original version here.

7.3 Some Properties of Integrals   - Reading: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Section 7.3: Some Properties of Integrals” Link: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Section 7.3: Some Properties of Integrals” (PDF)

Instructions: Please click on the link above and read Section 7.3 (pages 156-160) in its entirety.  In particular, note that the definite integral enjoys the same linearity properties that the derivative does, in addition to some others.  In its application to velocity functions, pay particular attention to the distinction between distance traveled and net distance traveled.

This reading should take you approximately one hour to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  This text was originally written by Professor David Guichard.  It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills.  You can access the original version here.

• Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 15: Antiderivatives” Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 15: Antiderivatives” (YouTube)

Instructions: Please click on the link above and watch the video from the beginning to the 30:00 minute mark.  Lecture notes are available here.

Viewing this lecture and taking notes should take you approximately 45 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  It is attributed to David Jerison and MIT's OpenCourseWare.  It may be viewed in its original form here.

• Assignment: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Exercises 7.3: Problems 1-6” Link: Whitman College: David Guichard’s Calculus: Chapter 7: Integration: “Exercises 7.3: Problems 1-6” (PDF)

Instructions: Please click on the link above link and work through problems 1-6 for Exercises 7.3.  When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately 45 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  This text was originally written by Professor David Guichard.  It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills.  You can access the original version here.

7.4 Integration by Substitution   - Reading: Whitman College: David Guichard’s Calculus: Chapter 8: Techniques of Integration: “Section 8.1: Substitution” Link: Whitman College: David Guichard’s Calculus: Chapter 8: Techniques of Integration: “Section 8.1: Substitution” (PDF)

Instructions: Please click on the link above and read Section 8.1 (pages 161-166) in its entirety.  This section explains the process of taking the integral of slightly more complicated functions.  We do this by implementing a “change of variables,” or rewriting a complicated integral in terms of elementary functions that we already know how to integrate.  Simply put, integration by substitution is merely the act of taking the chain rule in reverse.

This reading should take approximately one hour and 30 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  This text was originally written by Professor David Guichard.  It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills.  You can access the original version here.

• Lecture: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 15: Antiderivatives” Link: Massachusetts Institute of Technology: David Jerison’s Single Variable Calculus: “Lecture 15: Antiderivatives” (YouTube)

Instructions: Please click on the link above and watch the video from the 30:00 minute mark to the end.  Lecture notes are available here.

Viewing this lecture and taking notes should take you approximately 45 minutes to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  It is attributed to David Jerison and MIT's OpenCourseWare.  It may be viewed in its original form here.

• Assignment: Whitman College: David Guichard’s Calculus: Chapter 8: Techniques of Integration: “Exercises 8.1: Problems 5-19” Link: Whitman College: David Guichard’s Calculus: Chapter 8: Techniques of Integration: “Exercises 8.1: Problems 5-19” (PDF)

Instructions: Please click on the link above link and work through problems 5-19 for Exercises 8.1.  When you are done, check your answers against “Appendix A: Answers”.

This assignment should take you approximately two hours to complete.

Terms of Use: This resource is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 3.0 License.  This text was originally written by Professor David Guichard.  It has since been modified to include edited material from Neal Koblitz of the University of Washington, H. Jerome Keisler of the University of Wisconsin, Albert Schueller, Barry Balof, and Mike Wills.  You can access the original version here.