**Unit 2: Riemann Integration in Higher Dimensions**
*As you learned in* MA241*, the
integral is a form of a continuous sum that is developed and used to
measure the area between the graph of a function and the X-axis. The
graph of a “nice-enough” function
encloses an area in
which, at least intuitively, has volume. The Riemann integral for such
functions turns this idea into a concrete definition that is a direct
generalization of the concept of the Riemann integral for functions
.
Unlike the case of differentiability, which we explored in the previous
unit of this course, the passage to higher dimensions presents
relatively little difficulty in the case of the Riemann integral. In
this unit you will define the n-dimensional Riemann integral; study the
basic properties of this integral; see the subtleties that do arise for
this case; and learn two important computation techniques: repeated
integrals and the Change of Variables formula. The latter is a
generalization of the Change of Variable formula that you learned about
in MA241.*

**Unit 2 Time Advisory**

Completing this unit should take you approximately 45 hours.

☐ Subunit 2.1: 3 hours

☐ Subunit 2.2: 8 hours

☐ Subunit 2.3: 34 hours

**Unit2 Learning Outcomes**

Upon successful completion of this unit, you will be able to:
- define the Riemann integral of a function
;

- define and recognize sets of zero content;

- state and prove the basic properties of the Riemann integral;

- identify Riemann integrable functions;

- relate multiple and iterated integrals;

- compute Riemann integrals using iterated integrals;

- deine and recognize Jordan measurable sets;

- summarize the role of Jordan measurable sets in the theory of the Riemann integral;

- formulate and prove the Change of Variables formula;

- compute integrals with the Change of Variables formula; and

- use polar and spherical coordinates.

**2.1 A Review of Riemann Integration in
**
- **Reading: Trinity University: Dr. William F. Trench's Introduction
to Real Analysis: “Chapter 3, Section 1: Definition of the
Integral” and “Chapter 3, Section 2: Existence of the Integral”**
Link: Trinity University: Dr. William F. Trench's

*Introduction to Real Analysis*: “Chapter 3, Section 1: Definition of the Integral” (PDF) and “Chapter 3, Section 2: Existence of the Integral” (PDF)

Instructions: Read Chapter 3, titled “Integral Calculus of Functions of One Variable,” on pages 113 through 134. Skip the subsection on page 125 regarding the Riemann-Stieltjes Integral. You will return to this textbook throughout this unit, so you may prefer to save this PDF to your computer for quick reference.

The aim of this reading is to refresh your memory regarding the definition and intuitive geometric meaning of the integral of functions of one variable. As you read, please pay particular attention to the notions of intervals, lower and upper sums, and the rigorous definition of the Riemann integral and its geometric interpretation. Remember that this reading comprises a review of material you already have learned, and thus you do not need to complete the reading’s supplementary exercises unless you feel they will benefit your studies. Please keep in mind that you may find yourself returning to this reading to review material as you approach the remaining topics of this unit.

Taking this review reading seriously will provide you with a sound foundation for understanding the material in the rest of this unit. Unless you feel completely fluent with the details of the Riemann integral from MA241, you should spend up to 3 hours on this reading.

Terms of Use: Please respect the terms of use and copyright displayed on the webpage above.

**2.2 Riemann Integration in R ^{n}**

**2.2.1 Integrals over Rectangles**-

**Reading: Trinity University: Dr. William F. Trench’s**Link: Trinity University: Dr. William F. Trench’s

*Introduction to Real Analysis*: “Chapter 7, Section 1: Definition and Existence of the Multiple Integral”*Introduction to Real Analysis*: “Chapter 7, Section 1: Definition and Existence of the Multiple Integral” (PDF)

Instructions: Read Section 1, “Definition and Existence of the Multiple Integral.” Begin on page 435 at section titled “Integrals over Rectangles” and read through page 441, stopping when you reach the section titled “Upper and Lower Integrals.”

As you read through the material, visualize only the case n=2, as this case is sufficient to understand the subtleties of all higher dimensions. Try to absorb the concepts and results you see on these pages as directly analogous to familiar concepts from MA241. In particular, notice how the concepts of intervals and the Riemann integral that you reviewed in Subunit 2.1 are now being generalized to higher dimensions.

Reading this section should take approximately 1 hour.

Terms of Use: Please respect the terms of use and copyright displayed on the webpage above.

**2.2.2 The Volume of Rectangles; Upper and Lower Sums**
- **Reading: Trinity University: Dr. William F. Trench’s Introduction
to Real Analysis: “Chapter 7, Section 1: Definition and Existence
of the Multiple Integral”**
Link: Trinity University: Dr. William F. Trench’s

*Introduction to Real Analysis*: “Chapter 7, Section 1: Definition and Existence of the Multiple Integral” (PDF)

Instructions: Read Section 1, titled “Definition and Existence of the Multiple Integral.” Start on page 441 at the section titled “Upper and Lower Integrals.” Read through page 448 and stop at the section titled “Sets with Zero Content.”

In this reading you will encounter many elementary properties of the Riemann integral in higher dimensions. To help you interpret these results in the right context, please note that the text refers to results about the Riemann integral of functions of one variable, which you reviewed at the beginning of this unit. In this sense, the new results about the Riemann integral in higher dimensions are analogous to familiar properties of the Riemann integral from MA241. Note that many of the proofs presented in these pages are left to you, the reader, to complete. Please be sure to work through these exercises as you read.

Reading this section and completing the exercises should take approximately 3 hours.

Terms of Use: Please respect the terms of use and copyright displayed on the webpage above.

**2.2.3 Sets of Zero Content**
- **Reading: Trinity University: Dr. William F. Trench’s Introduction
to Real Analysis: “Chapter 7, Section 1: Definition and Existence
of the Multiple Integral”**
Link: Trinity University: Dr. William F. Trench’s

*Introduction to Real Analysis*: “Chapter 7, Section 1: Definition and Existence of the Multiple Integral” (PDF)

Instructions: Read Section 1, entitled “Definition and Existence of the Multiple Integral.” Start on page 448 at the section titled “Sets of Zero Content.” Read through page 453 and stop when you reach the section titled “Differentiable Surfaces.”

In this reading, you will encounter the geometric subtleties of higher dimensions that relate to integration. Intuitively, sets of zero content are sets that do not affect the Riemann integral, and thus these sets are important in stating properties of the Riemann integral and understanding its behavior. Sets of zero content also are relevant for the Riemann integral of functions of one variable. However, in higher dimensions, sets of zero content can be very complicated and include many interesting cases. You will encounter some of these sets in the next sub-subunit of this course.

Reading this material should take approximately 1 hour and 30 minutes.

Terms of Use: Please respect the terms of use and copyright displayed on the webpage above.

**2.2.4 Differentiable Surfaces**
- *Introduction
to Real Analysis*: “Chapter 7, Section 1: Definition and Existence
of the Multiple Integral”*Introduction to
Real Analysis*: “Chapter 7, Section 1: Definition and Existence of
the Multiple
Integral”
(PDF)

Instructions: Read Section 1, “Definition and Existence of the
Multiple Integral.” Start on page 453 at the section titled
“Differentiable Surfaces.” Read through page 455 and stop when you
reach the section titled “Properties of Multiple Integrals.”

This chapter presents a particular family of sets of zero content.
These sets are carved out of the ambient space ** R^{n}**
by the use of differentiable equations. You will see in the examples
presented in the text that such sets include many familiar and
important spaces.

Reading this section should take approximately 1 hour.

Terms of Use: Please respect the terms of use and copyright displayed on the webpage above.

**2.2.5 Properties of the Riemann Integrals**
- *Introduction
to Real Analysis*: “Chapter 7, Section 1: Definition and Existence
of the Multiple Integral”*Introduction to
Real Analysis*: “Chapter 7, Section 1: Definition and Existence of
the Multiple
Integral”
(PDF)

Instructions: Read Section 1, entitled “Definition and Existence of
the Multiple Integral.” Start on page 455 at the section titled
“Properties of Multiple Integrals.” Read through page 459 and stop
when you reach the section entitled “7.1 Exercises.”

In this reading, you will explore the many basic properties that
are true of the Riemann integral, which was developed at the
beginning of this unit. Notice that the results presented here in
the text are direct analogues of the properties of the Riemann
integral for functions of one variable. Also note that most of the
proofs presented on these pages are left as exercises for you, the
reader, to complete. It is vital that you complete the proofs
presented here in order to cement your understanding of these
concepts. If you’ve carefully studied the material developed so far
in this course, especially in the context of the familiar results of
MA241, then you should be
able to complete all the proofs presented on these pages.

Reading this material should take approximately 1 hour and 30
minutes.

Terms of Use: Please respect the terms of use and copyright
displayed on the webpage above.

**2.3 Techniques of Integration**
**2.3.1 Preparatory Exercises**
- **Activity: Trinity University: Dr. William F. Trench’s
Introduction to Real Analysis: “Chapter 7, Section 1: Definition
and Existence of the Multiple Integral”**
Link: Trinity University: Dr. William F. Trench’s

*Introduction to Real Analysis*: “Chapter 7, Section 1: Definition and Existence of the Multiple Integral” (PDF)

Instructions: Work through all the exercises in the section titled “7.1 Exercises” beginning on page 459.

So far, this unit of the course has covered the definition and basic properties of the Riemann integral in higher dimensions. This set of exercises is concerned with developing techniques of computation and simplification of the Riemann integral. You should proceed to the next sub-subunit of this course only after you feel comfortable with the exercises presented on these pages.

Working through these exercises should take approximately 10 hours.

Terms of Use: Please respect the terms of use and copyright displayed on the webpage above.

**2.3.2 Iterated Integrals and Repeated Integrals**
- **Reading: Trinity University: Dr. William F. Trench's Introduction
to Real Analysis: “Chapter 7, Section 2: Iterated Integrals and
Multiple Integrals”**
Link: Trinity University: Dr. William F. Trench's

*Introduction to Real Analysis*: “Chapter 7, Section 2: Iterated Integrals and Multiple Integrals” (PDF)

Instructions: Read Section 2: “Iterated Integrals and Multiple Integrals” on pages 462 through 484.

As you read this section, you will be introduced to a very important technique for reducing the computation of a Riemann integral in dimension

*n*to the computation of n ordinary Riemann integrals. This new technique allows all the techniques for computations of integrals that you learned in MA241 to be used in higher dimensions as well. This section is quite lengthy, not because the technique is very complicated, but rather because many examples are worked out in great detail. You will see that in practice, this is a rather simple method to implement. Be sure to work through all the problem exercises presented on pages 480-484 of this reading.

Reading this section and completing the exercises should take approximately 10 hours.

Terms of Use: Please respect the terms of use and copyright displayed on the webpage above.

**2.3.3 The Change of Variables Formula**
*This sub-subunit is concerned with the Change of Variables formula for
the Riemann integral in higher dimensions. Recall that the Change of
Variable formula for the ordinary Riemann integral, which you met in*
MA241*, involved changing the
bounds of integration ***–*** in effect, changing the set over which the
function is integrated. A similar phenomenon occurs in higher dimensions
as well, but in higher dimensions the change required can be far more
elaborate and subtle than it is in the one-dimensional case. In the
following sub-subunits of this course, prepare yourself to be confronted
with rather complicated geometrical difficulties before arriving at the
sought formula.*

**2.3.3.1 Change of Variables in Multiple Integrals**
- **Reading: Trinity University: Dr. William F. Trench's Introduction
to Real Analysis: “Chapter 7, Section 3: Change of Variables in
Multiple Integrals”**
Link: Trinity University: Dr. William F. Trench's

*Introduction to Real Analysis*: “Chapter 7, Section 3: Change of Variables in Multiple Integrals” (PDF)

Instructions: Read Section 3, entitled “Change of Variables in Multiple Integrals,” on pages 484 through 494, and stop when you reach the section entitled “Formulation of the Rule for Change of Variables.”

This section will guide you through the concepts of Jordan measurable sets, transformations of such sets, and the change of content that occurs under a linear transformation. It turns out that Riemann integration works well when integrating over what are known as Jordan measurable sets. Anticipating that a change of variables will distort the set upon which the function is integrated, you first must understand how Jordan measurable sets are transformed – in particular, the effect of linear transformations on the content of such sets.

Reading this section should take approximately 1 hour.

Terms of Use: Please respect the terms of use and copyright displayed on the webpage above.

**2.3.3.2 Formulation of the Rule for Change of Variables**
- **Reading: Trinity University: Dr. William F. Trench's Introduction
to Real Analysis: “Chapter 7, Section 3: Change of Variables in
Multiple Integrals”**
Link: Trinity University: Dr. William F. Trench's

*Introduction to Real Analysis*: “Chapter 7, Section 3: Change of Variables in Multiple Integrals” (PDF)

Instructions: Read Section 3, entitled “Change of Variables in Multiple Integrals,” on page 494 through 496, and stop when you reach the section entitled “The Main Theorem.”

This section will guide you through the geometric intuition that leads to the Change of Variables formula. Since the formula itself takes some time to get used to, it would be a good idea for you to read through this section at least twice. On your first reading, try to absorb the geometric intuition, but do not stop to contemplate any details. Following your first reading, spend some time considering and interpreting the Change of Variables formula. Then, read these pages again, this time spending as much time as is required for you to understand each step.

Reading this section should take approximately 1 hour.

Terms of Use: Please respect the terms of use and copyright displayed on the webpage above.

**2.3.3.3 The Main Theorem**
- **Reading: Trinity University: Dr. William F. Trench's Introduction
to Real Analysis: “Chapter 7, Section 3: Change of Variables in
Multiple Integrals”**
Link: Trinity University: Dr. William F. Trench's

*Introduction to Real Analysis*: “Chapter 7, Section 3: Change of Variables in Multiple Integrals” (PDF)

Instructions: Read Section 3, entitled “Change of Variables in Multiple Integrals,” on page 496 through 505, and stop when you reach the section entitled “Polar Coordinates.”

This section is concerned with the proof of the Change of Variables formula. This proof is quite involved and certainly not easy to comprehend – it is therefore recommended that you tackle this proof in two stages. For this assignment, try not to spend more than one hour reading over the material, keeping in mind that it is acceptable not to understand each step of the proof at this point. You can refer back to this reading after you have progressed through the next subunit of this course and completed a few exercises that apply the Change of Variables formula. After you have worked through some exercises using this formula, you likely will find this proof more palatable and easier to understand.

Reading this section should take approximately 2 hours.

Terms of Use: Please respect the terms of use and copyright displayed on the webpage above.

**2.3.3.4 Polar Coordinates, Spherical Coordinates, and Exercises**
- *Introduction
to Real Analysis*: “Chapter 7, Section 3: Change of Variables in
Multiple Integrals”*Introduction to
Real Analysis*: “Chapter 7, Section 3: Change of Variables in
Multiple
Integrals”
(PDF)

Instructions: Read Section 3, entitled “Change of Variables in
Multiple Integrals,” on page 505 through 517.

In this reading, you will see the Change of Variables formula
demonstrated for two particular changes of variables known as polar
coordinates and spherical coordinates. These changes of variables
have a geometric interpretation – as explained and illustrated in
the text – so as to make it easier for you to understand the general
formula. Be sure to work through all the problem exercises presented
on pages 514-517 of this reading.

Reading this section and completing the exercises should take
approximately 10 hours.

Terms of Use: Please respect the terms of use and copyright
displayed on the webpage above.