Lecture Note
Notes on Double Integrals
-
University:
Massachusetts Institute of Technology -
Course:
18.024 | Multivariable Calculus with Theory Academic year:
2022
-
Views:
249
Pages:
27
Author:
Rebecca Burton
Notes on double integrals.
(Read 11.1-11.5 of Apostol.)
Just as for the case of a single integral, we have the
following condition for the existence of a double integral:
is defined on
f is integrable on Q if and only if
-are step functions s and t with
-
Theorem 1 (Riemann condition).
.
Q = [arb] x [c,d]
given any
s
>
E
Then
there
0,
Suppose
f
< f
0,
Then
D
is
there i s a f i n i t e
and t h e sum of whose
E.
Examples.
f i n i t e s e t h a s c o n t e n t zero.
(I)
A
(2)
A h o r i z o n t a l l i n e segment h a s c o n t e n t zero.
(3)
A v e r t i c a l l i n e segment h a s c o n t e n t zero.
(4)
A s u b s e t o f a set o f c o n t e n t z e r o h a s c o n t e n t zero.
(5) A f i n i t e union o f sets o f c o n t e n t z e r o h a s c o n t e n t zero.
(6) The graph o f a c o n t i n u o u s f u n c t i o n
y = $(x);
i
h a s c o n t e n t zero.
a
< x <.b (7)
The graph of a continuous function
x=$(y);
c G y 9 d
has c o n t e n t zero.
Most of t h e s e statements a r e t r i v i a l t o prove: only t h e
l a s t t w o r e q u i r e some c a r e .
L e t us prove ( 6 ) .
Given t h e continuous function
,
Let
E'
> 0.
l e t us use t h e small-
span theorem f o r functions of a s i n g l e v a r i a b l e t o choose a
a = xo < x l <
partition
span of
$J
... < xn. = b
of
[a ,b]
on each s u b i n t e r v a l i s less than
E'.
such t h a t t h e
Consider t h e
rectangles
for
1
( 1
i = l,...,n.
-
Qx
i
They cover t h e graph of
1 <
E
whenever
The t o t a l a r e a of t h e r e c t a n g l e s
i=l
x
Q,
because
i s i n the i n t e r v a l
Ai
( x ~ - i-1
x )2€' = 2c1(b
equals
- a).
,xi]
. T h i s number e q u a l s
E
= e / 2 (b-a)
E
i f w e begin t h e proof by s e t t i n g
.
We now prove an elementary f a c t about sets of c o n t e n t zero:
Let Q be-a r e c t a n g l e . Let D be-a s u b s e t of
Q -t h a t has c o n t e n t zero. Given E > 0 , t h e r e i s-a p a r t i t i o n of
such t h a t t h o s e s u b r e c t a n g l e s of the partition t h a t contain
Q -o f D have
points a r e a l e s s t h a n E.
- t o t a l --Lemma 5.
Note t h a t t h i s lemma does n o t s t a t e merely t h a t
is
D
contained i n t h e union o f f i n i t e l y many s u b r e c t a n g l e s of t h e part i t i o n having t o t a l a r e a l e s s t h a n
E,
b u t t h a t t h e sum of t h e
areas of a l l t h e s u b r e c t a n g l e s t h a t c o n t a i n p o i n t s of
i
than
E.
D
is l e s s
The f o l l o w i n g f i g u r e i l l u s t r a t e s t h e d i s t i n c t i o n ;
D
i s c o n t a i n e d i n t h e union of two s u b r e c t a n g l e s , b u t t h e r e a r e
seven s u b r e c t a n g l e s t h a t c o n t a i n p o i n t s of
Proof.
A1,...,An
F i r s t , choose f i n i t e l y many r e c t a n g l e s
of t o t a l a r e a l e s s t h a n
"Expandm each one s l i g h t l y .
,J
rectangle
A;
Ai
D.
whose union c o n t a i n s
c/2
That i s , f o r each
whoseinterior contains
i s no more t h a n t w i c e t h a t o f
Ai.
Ai,
i,
D.
choose a
such t h a t t h e a r e a o f
Then the union o f t h e sets
Int
contains
A i
area l e s s than
outside
Q,
Of
so l e t
i n t e r s e c t i o n of
A:
and t h e r e c t a n g l e s
D,
have t o t a l
A;
course, the rectangle
may extend
denote t h e r e c t a n g l e t h a t i s t h e
A:
and
Then t h e r e c t a n g l e s
Q.
have t o t a l a r e a l e s s t h a n
also
A:
E.
Now use t h e end p o i n t s of t h e component i n t e r v a l s of t h e
rectangles
AT
t o define a partition
P
of t h e r e c t a n g l e
Q.
See t h e f i g u r e .
W e show t h a t t h i s i s o u r d e s i r e d p a r t i t i o n .
Note t h a t by c o n s t r u c t i o n , t h e r e c t a n g l e
by
P,
s o t h a t it i s a union of s u b r e c t a n g l e s
Now i f a s u b r e c t a n g l e
it c o n t a i n s a p o i n t o f
lies in
%
and hence i n
of a l l t h e subrectangles
A
Int
Qij
A1;
Ai;.
Qij
is parti,tioned
Aj;
of
Qij
c o n t a i n s a p o i n t of
f o r some
kt
Suppose w e l e t
A
A
D,
then
s o that it a c t u a l l y
B
d e n o t e t h e union
t h a t c o n t a i n p o i n t s of
be the union of t h e r e c t a n g l e s
P.
Then
D;
B C A.
and l e t It follows that
1
area Qij
Q~j~~
NOW
L
Q~j~~ area Qij*
on the other hand, by additivity of area for rectangles,
It follows that
This last inequality is in general strict, because some subso
rectangles i' j belong to more than one rectangle ,
their areas are counted more than once in the sum on the right
side of the inequality.
It follows that
as desired. 0
Now
we prove our basic theorem on existence of the double
If f is bounded on Q, and is continuous
f exists.
except on
zero, then
- -a- set
- of content Theorem 6.
on
-
Q
I/', Proof. S t e ~
1. We prove a preliminary result:
Suppose that given e > 0, there exist functions g and h that are integrable over Q, such
that
g(x) I f(x) l h(x)
for x in Q
and
Then f is integrable over Q.
We prove this result as follows: Because h and g are integrable, we can find step
functions sl, s2, tl, t2 such that
s1 1g 5 tl and s2 5 h j t2,
and such that
Consider the step functions sl and t2. We know that
s1Iglf1h1t2
so sl is beneath f, and t2 is above f. Furthermore, because the integral of g is between
the integrals of sl and of tl, we know that
Similarly,
If we add these inequalities and the inequality
we have
Since e is arbitrary, the 'kiemann condition is satisfied, so f is integrable over Q. S t e ~2. Now we prove the theorem. Let D be a set of zero content containing the
discontinuities of f. Choose M so that If(x) 1 5 M for x in Q; then given r > 0, set r t =
r/2M. Choose a partition P of Q such that those subrectangles that contain points of D
have total area less than
6).
(Here we use the preceding lemma.)
Now we define functions g and h such that g
< f < h on Q. If Q..13 is one of the
subrectangles that does not contain a point of D, set
g(x) = f(x) = h(x)
for x E Q. .. Do this for each such subrectangle. Then for any other x in Q, set
1J
g(x) = -M
and h(x) = M.
ThengSfShonQ.
Now g is integrable over each subrectangle Q.. that does not contain a point of D,
1J
since it equals the continuous function f there. And g is integrable over each subrectangle Q.. that does contain a point of D, because it is a step function on such a
1J
subrectangle. (It is constant on the interior of Q. ..) The additivity property of the
U
integral now implies that g is integrable over Q.
Similarly, h is integrable over Q. Using additivity, we compute the integral
JLQ
(h-g) =
1JJ
(h-g) = 2M
Qij
1(area Q~ that contain points of D) Thus the conditions of Step 1hold, and f is integrable over Q.
Theorem 7. Sup~osef & bounded on Q, and f eauals 0 except on a set D of content
zero. Then
-
JJQf exists and
~ Q U zero
~ S
Proof. We apply Step 2 of the preceding proof to the function f.
Choose M so that If(x)l 5 M for x in Q; given
E
> 0, set
E'
= E / ~ M Choose
.
a
partition P such that those subrectangles that contain points of D have total area less
than E'.
Define functions g and h as follows: If Q.. is one of the subrectangles that does not
1J
contain a point of D, set g(x) = f(x) = 0 and h(x) = f(x) = 0 on Q. .. Do this for each
1J
such subrectangle. For any other x in Q, set
g(x) = -M
and h(x) = M.
ThengSfshonQ.
I
Now g and h are step functions on Q, because they are constant on the interior of
each subrectangle Q. .. We compute
1J
JJQ
h =M
(1(area Q.. that contain points of D))
13
Similarly,
E, SO
that f is integrable over Q. Furthermore,
Since E is arbitrary,
Corollary 8.
set of content
--
a IJQf exists, and if g
g bounded function that eauds f except on a Proof.
\
We w r i t e
by h y p o t h e s i s , and
corollary.
Then
-
g
+ (g-f).
g = f
Now
is integrable
f
i s i n t e g r a b l e by t h e preceding
f
i s i n t e g r a b l e and
g
Double i n t e g r a l s extended o v e r more g e n e r a l r e g i o n s .
In t h i s section,
(Read s e c t i o n 1 1 . 1 2 of Apostol.)
[IS f
~ p o s t o ld e f i n e s
set
f o r a function
f
d e f i n e d on a bounded
but then he quickly r e s t r i c t s himself t o t h e s p e c i a l
S,
c a s e where
S
i s a r e g i o n of Types I o r 11.
We d i s c u s s h e r e
the general case.
F i r s t , w e prove t h e f o l l o w i n g b a s i c e x i s t e n c e theorem:
Theorem 9.
Bd S
of
-
I n t S,
f
?
equal
f
$JS f
Q
on
x0).
f
The f u n c t i o n
xl.
i s c o n t i n u o u s -a t each
-1
f
be a rectangle c o n t d h i n g
T
and l e t
S,
?
S,
Hence
equal
0
S.
As
outside
of t h e i n t e r i o r of
xo
xo,
and
f
usual.
S.
S
Then
(because
is continuous
xl
because it equals z e r o on an open b a l l
The only p o i n t s where
has c o n t e n t zero.
point
i s a l s o continuous a t each p o i n t
are p o i n t s of ths boundary of
1
If
-
J
M
m
,
E
Z
b
n
.
s
l
L
Y
U
l
l
i n an open b a l l about
of t h e e x t e r i o r of
about
bounded s e t i n t h e p l a n e .
exists.
i s continuous a t each p o i n t
it equals
at
then
-
P roof. Let
-
let
N
has content
-
L e t S be
- -a
z ero, andif
-
!f
Q
S,
can' fail t o b e continuous
and this set, by assumption,
exists.
El Note:
-
Adjoining or deleting boundary points of
changes the value of
value of
S
only on a set of content zerol so that
f
/Js f remains unchanged. Thus
)'IS
f =
/Irnt
£1
for instance.
Let us remark on a more general existence theorem than
that stated in Theorem 9.
if
~d
If
is a bounded set, and -
S
has content zero, and if f is continuous on Int S
*
except on a set D of content zero, then
f exists. For
S
-
in this case the discontinuities of the extended function f
lie in the union of the sets
Bd S
content zero because both
S
Bd
and
and
D
and this set has
D,
do.
There are more general existence theorems even than this,
but we shall not consider them.
NOW we note that the basic properties of the double
integral hold also for this extended integral:
Theorem 10.
Let
-
---
be
- -a bounded set in the plane.
S
has the following properties:
(a) Linearity,
the left side e x i s t s if the r i g h t s i d e does.
(b) Comparison.
If
-
f < g
provided both integrals e x i s t .
on t h e set
St
then
One
- (c) Additivity.
i
Let
S = SL
U
S2.
S1 n S2
If
has content
zero, then
provided the right side exists.
(a) Given
Proof.
ft
gr
cl
+
respectively, on
dg
equals cf
+
S and equal
dg on
S
and
0
0
S,
let
otherwise.
otherwise.
2, g equal
Then
Let
Q
We know that
be a rectangle containing S.
t
defined on
f, g
from this linearity follows.
(b) Similarly, if
f< g, then
,
from which we
conclude that
(c) Let
equal
on
S2,
f
Q
on S1,
be a rectangle containing
and equal
0
elsewhere. Let
and equal 0 elsewhere. Let
and equal 0
S.
f3 equal
elsewhere. Consider the function
Let fl
f2 equal
f on
Sf
f I
it equals
Because
zero.
f
on the set S1 n S2,
Sl n S2
and equals zero elsewhere.
has content zero,
l/
f4
Q
exists and equals
Now
linearity implies that
How
can one evaluate
iIS
f when
region? The computation is easy when
I or I1 and
evaluates
f
S
S
is a general
is a region of type
is continuous on the interior of
iterated integration.
1 367
4 f ofbyApostol.
S:
one
This result is
proved on p ,
using additivity, one can also evaluate
many other regions as well.
continuous function f
llS f
for
For example, to integrate a
over the region S pictured, one can break it up as indicated into
i
two
regions Sl and
intersect in a set of content zero.
and
and
S2
Since S1
S2 that
is of type I
is of type 11, we can compute the integrals
{I
s,
I[
f
s,A.
by iterated integration. We add the results to
f
obtain
Area.
We can now construct a rigorous theory of area.
We
already have defined the area of the rectangle Q = [a,b]
[c,d]
x
by the equation
area Q =
11
I..
Q
,I
We use this same equation for the general definition.
Definition.
L~-tS be a bounded s e t i n the plane.
is Jordan-measurable i f
Note that i f
t:y Theorem 9.
Sfs
S
We say t h a t
1 e x i s t s ; i n t h i s case, we define
Bd S has content zero, then
S
is Jordan-measurable, by
The converse a l s o holds; the proof is l e f t a s an exercise.
The area function has the following properties:
Theorem 11.
Let
and
S
Jordan-
T
be measurable sets
-in
-the
-A
plane.
(1) (Monotonicity).
(2)
and only if
-
(Positivity).
S
-
If
Area
S C T, then area S
-
< area T.
-
S 3 0, and equality holds if
has content zero. (3)
then
-
Su T
'
content zero,
-is-a- set
- of
(Additivity) If S n T
-
Jordan-measurable and
area(SuT) = area S + area T.
(4) Area S = Area (Int S) = Area (Su Bd S)
Proof.
Let
Q
.
be a rectangle containing
S
and
T.
C lT(x)
.
Let
Define
FT
is(x) = 1
for
x
= 0
for
x ft S.
E
S
similarly.
(1) If S
$ (x)
is contained in T, then
Then by the comparison theorem,
area
s
(2) Since
0 =
for all S.
~f
=
0
Ifs 11, /I,% j'b
1 =
=
0.
1 = area
= area T.
s,
has content zero, then
$1
S
by Corollary 7.
I.
< 1, we have by the comparison theorem,
11, 11,
S
L <
1=
$1 is =
Q
0, Conversely, suppose
>
Given
E
on
such that
Q
to which
1 = 0.
s
Then
/I
there must be a step function t
0,
t
I/
f/
Q
t <
Now if a subrectangle Qij
is a step function.
of this partition contains a point of
then the value of
t
is defined
2=
P be a partition relative
Let
E.
is = 0.
Q
S
in its interior,
on this subrectangle must be at least
~ h u sthese subrectangles have total area less than
E.
Now
1.
S
is contained in the union of these subrectangles (of total area
less than
e)
and the partition lines.
Thus
S
has content
zero.
(3) Because
I!,
1 and
$1T
1
exist and
content zero, it follows from additivity that
andequals
S n T has
$1
1 exists
SUT
$1
1 + $ $ 1.
S
T
(4) Since the part of
it has content zero.
S
not in
Int S
lies in
Then additivity implies that
area S = area(1nt S)
= area (Int S)
+
area (S
-
Int S)
.
A similar remark shows that
area ( S u Bd SJ = area (Int S )
= area (Int S )
+ area (Bd S)
.
Bd S, Remark.
1
Let
be a bounded s e t i n t h e plane.
S
d i r e c t way of d e f i n i n g t h e a r e a of
S,
i n t e g r a t i o n t h e o r y , i s a s follows:
Let
taining
P
a r e a of a l l s u b r e c t a n g l e s of
A(P)
of t h e numbers
S.
be a r e c t a n g l e con-
of
P
let
Q,
denote t h e t o t a l .
a(P)
t h a t a r e contained
-
Define t h e i n n e r a r e a of
a (P) , a s
-
P
and d e f i n e t h e o u t e r a r e a o f
.
Q
&
S
and
S,
denote t h e t o t a l a r e a of a l l s u b r e c t a n g l e s o f
contain points of
A(P)
without developing
S.
Given a p a r t i t i o n
let
A
P
be t h e supremum
ranges o v e r a l l p a r t i t i o n s o f
S
Q;
t o b e t h e infemum of t h e numbers
If t h e i n n e r a r e a and o u t e r a r e a of
-
that
common v a l u e i s c a l l e d t h e a r e a of
S
a r e equal, t h e i r
S.
W e leave it a s a ( n o t t o o d i f f i c u l t ) e x e r c i s e t o show t h a t
t h i s d e f i n i t i o n o f a r e a i s t h e same as t h e one we have given.
Remark.
There i s j u s t one f a c t t h a t remains t o be proved
about o u r n o t i o n of a r e a .
W e w o u l d c e r t a i n l y w i s h it t o be t r u e
t h a t i f two s e t s S and T i n t h e p l a n e a r e "congruent" i n t h e s e n s e
1
of elementary geometry, t h e n t h e i r a r e a s a r e t h e same.
This f a c t i s i
n o t immediate from t h e d e f i n i t i o n of a r e a , f o r we used r e c t a n g l e s
w i t h s i d e s p a r a l l e l t o t h e c o o r d i n a t e axes t o form t h e p a r t i t i o n s
on which we based o u r n o t i o n of " i n t e g r a l " , and hence o f " a r e a " .
~t i s n o t immediate, for i n s t a n c e , t h a t t h e r e c t a n g l e s
p i c t u r e d below have t h e same a r e a , f o r t h e a r e a of
by approximating
sides.
ing
T
T
T
S
and
i s defined
by r e c t a n g l e s w i t h v e r t i c a l and h o r i z o n t a l
[Of c o u r s e , w e can w r i t e e q u a t i o n s f o r t h e c u r v e s boundand compute i t s a r e a by i n t e g r a t i o n , i f w e wish.]
Proof o f t h e i n v a r i a n c e o f a r e a under "congruence" w i l l
have t o w a i t u n t i l we s t u d y t h e problem o f change of v a r i a b l e s
i n a double i n t e g r a l .
T Eh erci ses
I
1.
[Hint:
Chwse
s and
t
that
so that
Q
determined by
2. ( a )
S CQ.
Bd S
S&
Since
IS
hz-s content zero.
,
exists
Let
S
and
SnT,
T
S
and
there a r e
P
of
functions
such
Q,
Show t h a t t h e subrectangles
.]
hzve t o t a l volume l e s s than
.
R~
be bounded subsets of
Give
S
.
< E
tk.at contain points of
P
(b) Show t h a t i f
and
[$ ( t - s )
Q and
Bd (S U T ) C (Bd SVBd T ) .
SVT
e x i s t s , then
t h a t a r e s t e p functions r e l a t i v e t o a p a r t i t i o n
<, Is jt o
s
ISS 1
Show t h a t i f
Show t h a t
.
an example where e q u a l i t y does not hold
T
a r e Jordan-measurable,
then s o a r e
and furthermore
ar.ea(SbT)
=
+ areaT
areas
-
area ( S n T ) .
Express in terms o f iterated integrals the double integral
3.
2 2
x y ,
where
is t h e b o u n d e d p o r t i o n o f t h e f i r s t
S
i
quadrant lying between the curves
lines
4.
.
y
= x
and
y = 4x.
(Do
xy
= 1
xy-plane,
= 2
xy
and the
not e v a l u a t e t h e i n t e g r a l s . )
A s o l i d is b o u n d e d a b o v e b y t h e s u r f a c e
by the
and
and by the plane
x = 2.
z = x
2
-
y
2
,
below
Make a sketch;
e x p r e s s i t s v o l u m e a s a n i n t e g r a l ; a n d f i n d t h e volume.
5.
Express i n t e r n of i t e r a t e d i n t e g r a l s t h e volume of t h e region
i n t h e f i r s t a t a n t of
z = 0
wd
R~ bcunded by:
x + 2y + z = 1.
(a) The s u r f a c e r
( b ) The s u r f a c e s
z = xy
and
z = xy
and
z = 0 and Let Q denote the rectangle [0,1] x [0,1] in the following exercises.
@(a) Let f(x,y) = l/(y-x)
f(x,y) = 0
Does
JJQf
if x # y,
if x = y.
exist?
(b) Let g(x,y) = sin (l/(y-x))
Does
if x # y,
JJQg exist?
= 1if x = 112 and y is rational,
@ ~ e f(x,y)
t
,-
Show that
f(x,y) = 0 otherwise
1
f exists but
f(x,y)dy fails to exist when x = 112.
0
JJQ
J
I
@ Let f(x,y) = 1if (x,y) has the form (a/p,b/p),
where a and b are integers and p is prime,
f(x,y) = 0 otherwise.
Show that
J J f(x,y)dy dx exists but
f does not.
JJQ
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