Department

MathematicsCourse Code

MAT237Y1Professor

John BlandStudy Guide

FinalThis

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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

whenever

1

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2

•S⊂R2is a bounded set whose boundary consists of one or more smooth curves,

and

•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

S⊂R3is a union of smooth surfaces and fis continuous on ¯

S.

2.2.2. main theoretical points. You are responsible for knowing1

•the deﬁnition of the integral ZZR

fdA

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

cZb

a

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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3

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

ZZR

f dA =Zb

aZd

c

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:S→Ris 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:a≤x≤b, φ(x)≤y≤ψ(x)}

for some continuous functions φ, ψ : [a, b]→R, then

ZZS

f dA =Zb

a Zψ(x)

φ(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:c≤y≤y, φ(y)≤x≤ψ(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)∈S⇐⇒ a≤x≤b

φ(x)≤y≤ψ(x)

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 S⊂R3, to write iterated integrals over S, it is necessary to translate

description of Sinto inequalities

(x, y, z)∈S⇐⇒

a≤x≤b

φ(x)≤y≤ψ(x)

Φ(x, y)≤z≤Ψ(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

Z1

0Z1

y

ex2dx dy.

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