Department

MathematicsCourse Code

MAT237Y1Professor

John BlandStudy Guide

FinalThis

**preview**shows pages 1-2. to view the full**7 pages of the document.**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

S

that 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

R

2

is that it makes sense to talk about integrals

of the form

whenever

∫∫

S

f

dA

1

Only pages 1-2 are available for preview. Some parts have been intentionally blurred.

2

∫∫∫

•

⊂

∫∫

×

fdA

S

R

2

is a bounded set whose boundary consists of one or more smooth curves,

and

•

f

is

con

tin

uous

on

S

¯

(whic

h

implies

in

particular

that

f

is

b

ounded.)

.

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

S

f

dV

if

S

⊂

R

3

is

a

union

of

smo

oth

surfaces

and

f

is

con

tin

uous

on

S

¯

.

2.2.2.

main theoretical points.

You are responsible for knowing1

•

the definition of the integral

∫∫

where

R

is a rectangle in

R

2

and

f

is 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

f

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

f

to 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

R

2

, then under suitable

hypotheses,

∫∫

f

dA

=

∫

d

.∫

b

f

(

x, y

)

dx

Σ

dy

R c a

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.

R

•

•

•

•

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