# MA102: Single-Variable Calculus II

Unit 1: The Integral   We will begin by quickly reviewing the basics of integration, so integration is fresh in your mind before we extend its applications.  Having completed MA101, you should be familiar with this material. We will then take a look at how integration applies to concepts like motion.  Finally, we will discuss how logarithmic and exponential functions are integrated.

This unit should take you approximately 20.25 hours to complete.

☐    Subunit 1.1: 14 hours

☐    Sub-subunit 1.1.1: 2 hours

☐    Sub-subunit 1.1.2: 2 hours

☐    Sub-subunit 1.1.3: 1.5 hours

☐    Sub-subunit 1.1.4: 4 hours

☐    Sub-subunit 1.1.5: 2.5 hours

☐    Sub-subunit 1.1.6: 2 hours

☐    Subunit 1.2: 6.25 hours

☐    Sub-subunit 1.2.1: 2.5 hours

☐    Sub-subunit 1.2.2: 1.5 hours

☐    Sub-subunit 1.2.3: 2.25 hours

Unit1 Learning Outcomes
Upon successful completion of this unit, the student will be able to: - Define and describe the indefinite integral. - Compute elementary definite and indefinite integrals. - Explain the relationship between the area problem and the indefinite integral. - Use the midpoint rule to approximate the area under a curve. - State the fundamental theorem of calculus. - Use change of variables to compute more complicated integrals. - Integrate transcendental, logarithmic, and hyperbolic functions.

1.1 Review of Integration   1.1.1 The Indefinite Integral   - Reading: University of Michigan’s Scholarly Monograph Series: Wilfred Kaplan’s and Donald J. Lewis’s Calculus and Linear Algebra Vol.1: “4-1 Introduction” and “4-2 The Indefinite Integral” Link: University of Michigan’s Scholarly Monograph Series: Wilfred Kaplan and Donald J. Lewis’s Calculus and Linear Algebra Vol. 14-1 Introduction” (HTML)  and   4-2 The Indefinite Integral (HTML)

`````` Instructions: Please click on the links above, and read Sections
4-1 and 4-2 in their entirety.  Note that for Section 4-2, you will
need to click on the “next” link at the bottom of each page to

How does one “undo” differentiation?  There are a few
considerations.  We need to loosely define “elementary function” to
mean a function put together from rational and trigonometric
functions, exponentials, radicals, and so forth.  Although the
derivative of an elementary function always is elementary, there are
elementary functions with no elementary antiderivative.  Since the
derivative of a constant is 0, two functions that differ only by an
added constant term will have the same derivative, so
antiderivatives are never unique.  However, these limitations do not
forbid us from developing a strong theory of antidifferentiation,

Studying these readings should take approximately 30 minutes.

displayed on the webpage above.
``````
• Reading: University of Michigan’s Scholarly Monograph Series: Wilfred Kaplan and Donald J. Lewis’s Calculus and Linear Algebra Vol. 1: “4- 5 Basic Properties of the Indefinite Integral” and “4-6 Applications of Rules of Integration” Link: University of Michigan’s Scholarly Monograph Series: Wilfred Kaplan’s  and Donald J. Lewis’s Calculus and Linear Algebra Vol. 1:  “4-5 Basic Properties of the Indefinite Integral” (HTML) and  “4-6 Applications of Rules of Integration” (HTML)

Instructions:  Please click on the links above, and read Sections 4-5 and 4-6 in their entirety.  Use the “previous” and “next” links at the bottom of the page to navigate through each reading.  These readings discuss the “nuts and bolts” of finding antiderivatives.

Studying these reading should take approximately 30 minutes.

• Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck and Molly M. Cow’s “Indefinite Integrals” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck and Molly M. Cow’s  “Indefinite Integrals” (HTML)

Instructions: Click on the above link.  Then, click on the “Index” button.  Scroll down to “1. Integration,” and click button 104 (Indefinite Integrals).  Do problems 1-10.  Use the buttons in the module to check your answer or to move on to subsequent problems.  If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

1.1.2 The Area Problem and the Definite Integral   - Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.1: The Definite Integral” Link: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration:  “Section 4.1: The Definite Integral” (PDF)

`````` Instructions:  Please click on the link above, and read Section 4.1
in its entirety (pages 175 through 185).

This is a slightly different presentation of definite integrals and
area from the one chosen for MA101.  The author approaches limits
using the idea of rigorously defined *infinitesimals*, which you may
think of as infinitely small numbers.  These ultimately give a
fairly intuitive theory, but you will need some vocabulary, because
it is not the typical presentation.  An *infinitesimal* is a number
strictly between −a and a, for every positive real a.  The only real
infinitesimal is 0; more appear when we use the *hyperreals*, which
are a collection of numbers extending the reals.  There are non-real
hyperreals between any two reals: for example, if ε is a positive
infinitesimal, 5+ε is a non-real hyperreal between 5 and 6 (or even
5 and 5.00000001).  The *standard part* of a hyperreal is obtained
by “rounding to the nearest real,” so the standard parts of 5+ε and
5−ε are both 5, and the standard part of ε by itself is 0.  Division
by a nonzero infinitesimal gives an infinite hyperreal, a hyperreal
that is strictly larger than any real number.  The author is careful
in the text to confirm the calculations give *finite* hyperreals.
Infinite hyperreals have no standard part.  The sum of two
infinitesimals is always infinitesimal, as is the product of an
infinitesimal and any finite hyperreal number.

The *Transfer Principle* states that any mathematical statement
true of a function on the reals is also true of that function’s
natural extension to the hyperreals.  For example, the fact that
(x + y)<sup>2</sup> = x<sup>2</sup> + 2xy + y<sup>2</sup> holds for
all real x and y means it also holds for all hyperreal x and y.  The
proof of this principle is not important for our purposes, but the
principle itself is vital.  Finally, a *hyperinteger* is the integer
analog to a hyperreal.  However, the only finite hyperintegers are
the integers, so you may think of the hyperintegers as an extension
of the integers into infinity.

Studying this reading should take approximately 1 hour.

is attributed to H. Jerome Kiesler and the original version can be
found [here](http://www.math.wisc.edu/~keisler/calc.html) (PDF).
``````
• Lecture: MIT: David Jerison’s “Lecture 18: Definite Integrals” and University of Houston: Selwyn Hollis’s “Video Calculus: The Integral” Lecture Link: MIT: David Jerison’s  “Lecture 18: Definite Integrals” (YouTube)

Also Available in:
iTunes U

and University of Houston: Selwyn Hollis’s  “Video Calculus: The Integral” (QuickTime)

Instructions: Please watch the first video lecture in its entirety (47:14 minutes).  Note that lecture notes are available in PDF; the link is on the same page as the lecture.

If you desire a shorter presentation, choose the second video; click on the second link; then scroll down to Video 22: “The Integral.” Professor Jerison discusses the area problem and the cumulative sum problem and uses them to define the definite integral.

Studying one of these lectures should take approximately 1 hour.

1.1.3 Approximating Integrals   - Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: The Area under a Curve” Link: University of Houston: Selwyn Hollis’s  “Video Calculus: The Area under a Curve” (QuickTime)

`````` Instructions: Please click on the link; then scroll down to Video
21: “The Area under a Curve.”  Please watch the entire lecture. We
numerically in a later unit using more complicated methods.

Viewing this lecture and pausing to take notes should take
approximately 45 minutes.

displayed on the webpage above.
``````
• Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: James Palermo and Matthias Beck’s “Midpoint Rule” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: James Palermo and Matthias Beck’s  “Midpoint Rule” (HTML)

Instructions: Click on the above link.  Then, click on the “Index” button.  Scroll down to “1. Integration,” and click button 111 (Midpoint Rule).  Do problems 1-5.  If you do not fully understand the midpoint rule, click on the “Help” button for a quick refresher.  If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 45 minutes.

1.1.4 The Fundamental Theorem of Calculus   The Fundamental Theorem of Calculus is a two part statement relating definite integrals to derivatives and antiderivatives.  First, it says that integrating f(t) from a to x and then differentiating results in f(x).  Second, it says that definite integrals may be calculated via antiderivatives.  It is important, because first, it says the relationship between integration and differentiation is what it ought to be, and second, that we can evaluate definite integrals without always resorting to limits of Riemann sums.

• Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.2: The Fundamental Theorem of Calculus” Link: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.2: The Fundamental Theorem of Calculus” (PDF)

Instructions:  Please click on the link above, and read Section 4.2 in its entirety (pages 186 through 197).  This is a recap of the fundamental theorem of calculus, which relates the area problem and the definite integral to antiderivatives.

Studying this reading should take approximately 1 hour.

Terms of Use: The article above is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML).  It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).

• Lecture: MIT: David Jerison’s “Lecture 19: First Fundamental Theorem of Calculus” Link: MIT: David Jerison’s “Lecture 19: First Fundamental Theorem of Calculus” (YouTube)

Instructions: Please watch this video lecture in its entirety.  This lecture will cover sub-subunits 1.1.3-1.1.5.  Professor Jerison states the first fundamental theorem of calculus and uses it to calculate some integrals.  He then discusses some properties of the definite integral and the method of substitution for computing integrals.

Viewing this lecture and pausing to take notes should take approximately 1 hour.

Terms of Use: The video above is released under a Creative Commons Attribution-Share-Alike License 3.0 (HTML).  It is attributed to MIT and the original version can be found here (Flash or MP4).

• Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: The Fundamental Theorem of Calculus” Link: University of Houston: Selwyn Hollis’s “Video Calculus: The Fundamental Theorem of Calculus” (QuickTime)

Instructions: This video lecture is optional.  Please click on the link; then scroll down to Video 23: “The Fundamental Theorem of Calculus.”  This video explains the definition of a function via integration and the differentiation of such functions.

Viewing this lecture and pausing to take notes should take approximately 30 minutes.

• Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Aaron Robertson’s “Differentiation and the Fundamental Theorem” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Aaron Robertson’s “Differentiation and the Fundamental Theorem” (HTML)

Instructions: Click on the above link.  Then, click on the “Index” button.  Scroll down to “1. Integration,” and click button 114 (Differentiation and the Fundamental Theorem).  Do problems 4-20.  If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour and 30 minutes.

1.1.5 Elementary Integrals   - Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.3: The Indefinite Integral” Link: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.3: The Indefinite Integral” (PDF)

`````` Instructions:  Please click on the link above, and read Section 4.3
in its entirety (pages 198 through 207).  In this text, the author
chooses to present the indefinite integral after the definite
integral; however, this should not interfere with your understanding
of the chapter.  The section is extremely well-written; pay close
attention to the discussion of antiderivatives, theorem 3, the rules
of integration, and example 9.

Studying this reading should take approximately 1 hour.

is attributed to H. Jerome Kiesler and the original version can be
found [here](http://www.math.wisc.edu/~keisler/calc.html) (PDF).
``````
• Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: Antidifferentiation and Indefinite Integrals” Link: University of Houston: Selwyn Hollis’s “Video Calculus: Antidifferentiation and Indefinite Integrals” (QuickTime)

Instructions: Please click on the link scroll down to Video 24: “Antidifferentiation and Indefinite Integrals,” and view the entire lecture.

Viewing this lecture and pausing to take notes should take approximately 45 minutes.

• Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Daniel Russo’s “Definite Integrals” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Daniel Russo’s “Definite Integrals” (HTML)

Instructions: Click on the above link.  Then, click on the “Index” button.  Scroll down to “1. Integration,” and click button 108 (Definite Integrals).  Do problems 5-15.  These should be very easy.  If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 45 minutes.

1.1.6 Integration by Substitution (Change of Variables)   The derivative of f(g(x)), by the chain rule, is f’(g(x))*g’(x).  If presented with the latter expression as our integrand, we should get f(g(x)) back as the integral.  However, it can be difficult to determine what f and g are at a glance.  Substitution “cleans up” the integrand by hiding g(x) inside a new variable u, and combining g’(x) and dx into du.  It may also be used to turn a root into a power: let u equal the radical, solve for x, then differentiate, and plug the result in for dx.  We will later see other uses of substitution, but all are designed to rewrite the problem into a form where the method of integration is clear.

• Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.4: Integration by Change of Variables” Link: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 4: Integration: “Section 4.4: Integration by Change of Variables” (PDF)

Instructions: Please click on the link above, and read Section 4.4 in its entirety (pages 209 through 215).

Studying this reading should take approximately 30 minutes.

Terms of Use: The above book is released under a Creative Commons Attribution-NonCommercial-ShareAlike License 3.0 (HTML).  It is attributed to H. Jerome Kiesler and the original version can be found here (PDF).

• Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: Change of Variables (Substitution)” Link: University of Houston: Selwyn Hollis’s “Video Calculus: Change of Variables (Substitution)” (QuickTime)

Instructions: Please click on the link, scroll down to Video 25: “Change of Variables (Substitution),” and view the entire lecture.

Viewing this lecture and pausing to take notes should take approximately 30 minutes.

• Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Dan Reich’s “Substitution Methods” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Dan Reich’s “Substitution Methods” (HTML)

Instructions: Click on the above link.  Then, click on the “Index” button.  Scroll down to “1.  Integration,” and click button 109 (Substitution Methods).  Do at least problems 1-10.  If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

1.2 Integration of Transcendental Functions   A transcendental number is a number that is not the root of any integer polynomial.  A transcendental function, similarly, is a function that cannot be written using roots and the arithmetic found in polynomials.  We address exponential, logarithmic, and hyperbolic functions here, having covered the integration and differentiation of trigonometric functions previously.

1.2.1 Exponential Functions   - Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.3: Derivatives of Exponential Functions and the Number e” Link: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.3: Derivatives of Exponential Functions and the Number e” (PDF)

`````` Instructions: Please click on the link above, and read Section 8.3
in its entirety (pages 441 through 447).  This chapter recaps the
definition of the number e and the exponential function and its
behavior under differentiation and integration.

Studying this reading should take approximately 45 minutes.

is attributed to H. Jerome Kiesler and the original version can be
found [here](http://www.math.wisc.edu/~keisler/calc.html) (PDF).
``````
• Lecture: University of Houston: Selwyn Hollis’s “Video Calculus: The Natural Logarithmic Function” and “The Exponential Function” Link: University of Houston: Selwyn Hollis’s “Video Calculus: The Natural Logarithmic Function” (QuickTime) and “The Exponential Function” (QuickTime)

Instructions: These lectures will cover sub-subunits 1.2.1 and 1.2.2.  Please watch these two video lectures AFTER doing the readings for 1.2.1 and 1.2.2.  Click on the first link, scroll down to Video 31: “The Natural Logarithmic Function,” and watch the presentation through the 5th slide (marked 5 of 8).  Next, click on the second link above, and scroll down to Video 32: “The Exponential Function.”  Choose the format that is most appropriate for your Internet connection, and listen to the entire 21 minute video lecture.

The first short video gives one definition of the natural logarithm and derives all the properties of the natural log from that definition.  It does a number of examples of limits, curve sketching, differentiation, and integration using the natural log.  We will return to this video later to watch the last three slides.  The second video explains the number e, the exponential function and its derivative and antiderivative, curve sketching using the exponential function, and how to perform similar operations on power functions with other bases using the change of base formula.

Viewing these lectures and note-taking should take approximately 45 minutes.

• Assessment: University of California, Davis: Duane Kouba’s “The Integration of Exponential Functions: Problems 1-12” Link: University of California, Davis: Duane Kouba’s “The Integration of Exponential Functions: Problems 1-12” (HTML)

Instructions: Click on the link above and work through all of the assigned problems.  When you are done, check your solutions with the answers provided.

Completing this assessment should take approximately 1 hour.

1.2.2 Natural Logarithmic Functions   - Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.5: Natural Logarithms” Link: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.5: Natural Logarithms” (PDF)

`````` Instructions: Please click on the link above, and read Section 8.5
in its entirety (pages 454 through 459).  This chapter reintroduces
the natural logarithm (the logarithm with base e) and discusses its
derivative and antiderivative.  Recall that you can use these
properties of the natural log to extrapolate the same properties for
logarithms with arbitrary bases by using the change of base
formula.

Studying this reading should take approximately 30 minutes.

is attributed to H. Jerome Kiesler and the original version can be
found [here](http://www.math.wisc.edu/~keisler/calc.html) (PDF).
``````
• Assessment: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck’s “Logarithm, Definite Integrals” Link: Temple University: Gerardo Mendoza’s and Dan Reich’s Calculus on the Web: Calculus Book II: Matthias Beck’s “Logarithms, Definite Integrals” (HTML)

Instructions: Click on the link above.  Then, click on the “Index” button.  Scroll down to “3. Transcendental Functions,” and click button 137 (Logarithm, Definite Integrals).  Do problems 1-10.  If at any time a problem set seems too easy for you, feel free to move on.

Completing this assessment should take approximately 1 hour.

1.2.3 Hyperbolic Functions   - Reading: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.4: Some Uses of Exponential Functions” Link: University of Wisconsin: H. Jerome Keisler’s Elementary Calculus: Chapter 8: Exponential and Logarithmic Functions: “Section 8.4: Some Uses of Exponential Functions” (PDF)

`````` Instructions: Please click on the link above and read Section 8.4
in its entirety (pages 449 through 453).  In this chapter, you will
learn the definitions of the hyperbolic trig functions and how to
differentiate and integrate them.  The chapter also introduces the
concept of capital accumulation.

Studying this reading should take approximately 15-20 minutes.

is attributed to H. Jerome Kiesler and the original version can be
found [here](http://www.math.wisc.edu/~keisler/calc.html) (PDF).
``````

Instructions: Click on the links above and watch the videos.  The creator of the video pronounces “sinh” as “chingk.”  The more usual pronunciation is “sinch.”

Viewing these lectures and pausing to take notes should take approximately 30 minutes.