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John Bland
Study Guide

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University of Toronto
MAT237Y: Multivariable Calculus,
2017-8, Notes 10
1. Summary
1.1. topics. These notes summarize material covered in the first three weeks of lectures of
the Winter Term. These include
review of integration for functions of a single variable.
abstract theory of integration for functions of 2 or more variables.
how actually to compute integrals in practice: iterated integrals.
Our treatment of this material is generally similar to that in Sections 4.1 – 4.2 of Folland’s
Advanced Calculus.
2. Discussion of material
2.1. abstract theory of integration for functions of a single variable. The lectures
covered most of Section 4.1 of Folland, but omitting some details of proofs, and largely
skipping the Fundamental Theorem of Calculus (because we assume that you remember
the statement and how to use it, and we did not want to go into the proof.)
This was mostly review from MAT137, and the main points here all reappear in our
discussion of integration in higher dimensions.
To refresh your memory, you may want to review the nice series of videos from MAT137
that go over the definition of the integral in 1 dimension. (Search for “definition of integral
mat137” on youtube.)
One of the concepts that appears here and that you may not have seen in MAT137
is the notion of “zero content”. This is important for integration in higher dimensions,
particularly when we want to integrate over any set Sthat is not a rectangle. Then “zero
content” can help us determine whether or not this is possible. One reason we study zero
content in 1 dimension is to help us get a grip on the concept for higher dimensions, where
it is really useful.
2.2. Integration of functions of two or more variables.
2.2.1. main practical conclusions. For practical purposes, the most important conclusion
from the abstract theory of integration in R2is that it makes sense to talk about integrals
of the form ZZS
f dA

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SR2is a bounded set whose boundary consists of one or more smooth curves,
fis continuous on ¯
S(which implies in particular that fis bounded.).
These are far from the most general possible conditions, but in practice, this is normally
the kind of integration problem we are interested in, particularly in MAT237.
Similar considerations apply in higher dimensions, although we did not discuss everything
in complete detail. But for example in 3d, it is always possible to talk about RRRSf dV if
SR3is a union of smooth surfaces and fis continuous on ¯
2.2.2. main theoretical points. You are responsible for knowing1
the definition of the integral ZZR
where Ris a rectangle in R2and fis a function defined on R. This includes the
definitions of
upper and lower Riemann sums
upper and lower Roiemann integral
integrability, and (when a function is integrable) the integral RRf dA.
basic properties of integration; see Theorem 4.17 in Folland’s Advanced Calculus,
Section 4.2.
The definition of zero content, and basic properties of zero content, see Proposition
4.19 in Folland’s Advanced Calculus, Section 4.2.
The definition of a measurable set.
Sufficient conditions for a function fto be integrable on a set S. SeeTheorem
4.21 in Folland’s Advanced Calculus, Section 4.2. This is a more general and more
complicated version of the “main practical conclusion” summarized in Section 2.2.1
above of these notes.
2.3. Iterated integrals. In practice, one almost never computes a integral straight from
the definition involving Riemann sums. Instead, we reduce the problem of integration to a
problem of carrying out repeated 1-dimensional integrals, which we can do using techniques
from single-variable calculus.
The main fact is that if R= [a, b]×[c, d] is a rectangle in R2, then under suitable
hypotheses, ZZR
f dA =Zd
f(x, y)dxdy
1this means: first make sure that you know the basic material – the definitions and the main theorems
etc — then make sure that you are able to use this material to solve problems. Should you memorize proofs?
Not unless
you are completely sure that you know the basic material and can use the it to solve problems.
you understand the theorems well enough to
extract what you consider to be the main steps/main ideas of the proofs,
remember only these, and
reconstruct the proofs from your memory of the main steps.

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The integral on the right is called an iterated integral. The hypotheses are that every
integral appearing in the above formula (there are three of them) involves the integration
of an integrable function. Similarly, the identity
f dA =Zb
f(x, y)dydx
also holds under suitable hypotheses. See Theorem 4.26 in Folland’s Advanced Calculus,
Section 4.3 for details.
The hypotheses that we mentioned above are always satisfied if Sis a closed, measurable
set and f:SRis continuous.
The chief difficulty one encounters, in practice, arises when integrating over a set Sthat
is not a rectangle. If Shas the form
(1) S={(x, y)R2:axb, φ(x)yψ(x)}
for some continuous functions φ, ψ : [a, b]R, then
f dA =Zb
a Zψ(x)
f(x, y)dy!dx
Similarly it is straightforward to write down a suitable interated integral when the set Sis
described as S={(x, y)R2:cyy, φ(y)xψ(y)}.
The difficulty is that Sis ususally not described in a convenient form such as (1). So
the challenge one often ecounters is, given a set S, translate its description into a set of
inequalities, such as
(x, y)Saxb
or similar inequalities with the roles of xand yreversed. It may also sometimes be necessary
to split a set Sinto mutiple pieces, each one of which is expressed by inequalities of the
above form.
The same issues arise in higher dimensions, except that they become more difficult. For
example, given a set SR3, to write iterated integrals over S, it is necessary to translate
description of Sinto inequalities
(x, y, z)S
Φ(x, y)zΨ(x, y)
or similar inequalities with the roles of x, y, z permuted in some way.
A number of examples were considered in the lectures, and more examples can be found
in Section 4.3 of Folland.
An interesting class of examples involves iterated integrals that look impossible as writ-
ten, but that become possible once one exchanges the order of integration. Consider
ex2dx dy.
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