integral over plane region

integral over plane region

The integrals over a planar region are generalisations of usual

Riemann integrals

, but special cases of

http://planetmath.org/node/6660

surface integrals.

0.1

Integral over a rectangle

Let

R

be the

rectangle

of

x

⁢

y

-plane defined by

a

≦

x

≦

b

,

c

≦

y

≦

d

(1)

and the function

f

be defined and

bounded

in

R

.  Let

D

:

{

x

0

=

a

,

x

1

,

…

,

x

m

=

b

y

0

=

c

,

y

1

,

…

,

y

n

=

d

(2)

a of

R

into the rectangular parts

Δ

i

with areas

Δ

i

⁢

A

(

i

=

1

,

…

,

m

⁢

n

).  Denote

m

i

:=

inf

Δ

i

⁡

f

⁢

(

x

,

y

)

,

M

i

:=

sup

Δ

i

⁡

f

⁢

(

x

,

y

)

and

s

D

:=

∑

D

m

i

⁢

Δ

i

⁢

A

,

S

D

:=

∑

D

M

i

⁢

Δ

i

⁢

A

.

Definition 1.

If

sup

D

⁡

{

s

D

}

=

inf

D

⁡

{

S

D

}

,  then we say that

f

is

integrable

over

R

and call the common value the (

Riemann

)

integral

of

f

over the rectangle

R

and denote it by

∫

R

f

,

∫

R

f

⁢

(

x

,

y

)

⁢

𝑑

x

⁢

𝑑

y

or

∬

R

f

⁢

(

x

,

y

)

⁢

𝑑

x

⁢

𝑑

y

.

Let then

f

be defined in a region

A

of

x

⁢

y

-plane such that that it can be enclosed in a rectangle

R

defined by (1).  Define the new function

f

1

through

f

1

⁢

(

x

,

y

)

:=

{

f

⁢

(

x

,

y

)

when

⁢

(

x

,

y

)

∈

A

,

0

otherwise.

(3)

Definition 2.

If

f

1

is integrable over the rectangle

R

, we say that

f

is

integrable over

A

and define

∫

A

f

:=

∫

R

f

1

.

(4)

It’s apparent that (4) is on the choice of

R

since the points of

ℝ

2

∖

A

give zero-terms to the lower and upper sums.

0.2

Double integrals

Definition 3.

Let

f

be bounded in

R

as before.  Suppose that

φ

⁢

(

x

)

:=

∫

c

d

f

⁢

(

x

,

y

)

⁢

𝑑

y

is defined on

[

a

,

b

]

.  If also the integral

∫

a

b

φ

⁢

(

x

)

⁢

𝑑

x

=

∫

a

b

[

∫

c

d

f

⁢

(

x

,

y

)

⁢

𝑑

y

]

⁢

𝑑

x

(5)

exists, it is called a

double integral

or

iterated integral

and denoted by

∫

a

b

𝑑

x

⁢

∫

c

d

f

⁢

(

x

,

y

)

⁢

𝑑

y

.

One may prove the

Theorem.

∫

R

f

⁢

(

x

,

y

)

⁢

𝑑

x

⁢

𝑑

y

=

∫

a

b

𝑑

x

⁢

∫

c

d

f

⁢

(

x

,

y

)

⁢

𝑑

y

,  provided that the integral of the left side exists and that the inner integral

∫

c

d

f

⁢

(

x

,

y

)

⁢

𝑑

y

of the right side exists for every

x

in

[

a

,

b

]

.

It’s clear that

∫

a

b

𝑑

x

⁢

∫

c

d

f

⁢

(

x

,

y

)

⁢

𝑑

y

=

∫

c

d

𝑑

y

⁢

∫

a

b

f

⁢

(

x

,

y

)

⁢

𝑑

x

if also the integral

∫

a

b

f

⁢

(

x

,

y

)

⁢

𝑑

x

exists for every

y

in

[

c

,

d

]

.  If especially the function

f

is

continuous

in the rectangle

R

, then surely

∫

R

f

⁢

(

x

,

y

)

⁢

𝑑

x

⁢

𝑑

y

=

∫

a

b

𝑑

x

⁢

∫

c

d

f

⁢

(

x

,

y

)

⁢

𝑑

y

=

∫

c

d

𝑑

y

⁢

∫

a

b

f

⁢

(

x

,

y

)

⁢

𝑑

x

.

Assume now, that

f

is defined and bounded in the region

A

:=

{

(

x

,

y

)

∈

ℝ

2

⋮

a

≦

x

≦

b

,

h

1

(

x

)

≦

y

≦

h

2

(

x

)

}

where

A

is contained in the rectangle

R

determined by (1).  Then the planar integral

∫

A

f

is defined as

∫

A

f

=

∫

R

f

if the integral of right side exists.  For this, he

continuity

of

f

in

A

does not necessarily suffice, because

f

1

may have a

jump discontinuity

on the border of

A

whence the integrability of

f

1

needs not be guaranteed.  One case where the integrability is true is that the graphs of the functions

h

1

and

h

2

are

rectifiable

(i.e. the functions have continuous

derivatives

).  For a continuous

f

, we then have

∫

c

d

f

1

⁢

(

x

,

y

)

⁢

𝑑

y

=

∫

c

h

1

⁢

(

x

)

0

⁢

𝑑

y

+

∫

h

1

⁢

(

x

)

h

2

⁢

(

x

)

f

⁢

(

x

,

y

)

⁢

𝑑

y

+

∫

h

2

⁢

(

x

)

d

0

⁢

𝑑

y

=

∫

h

1

⁢

(

x

)

h

2

⁢

(

x

)

f

⁢

(

x

,

y

)

⁢

𝑑

y

.

Thus

∫

A

f

=

∫

R

f

1

=

∫

a

b

𝑑

x

⁢

∫

h

1

⁢

(

x

)

h

2

⁢

(

x

)

f

⁢

(

x

,

y

)

⁢

𝑑

y

,

(6)

i.e. the planar integral has been expressed as a double integral.

[Not ready …]

Title

integral over plane region

Canonical name

IntegralOverPlaneRegion

Date of creation

2013-03-22 18:50:57

Last modified on

2013-03-22 18:50:57

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

13

Author

pahio (2872)

Entry type

Definition

Classification

msc 26A42

Classification

msc 28-00

Synonym

planar integral

Related topic

RiemannMultipleIntegral

Defines

double integral

Defines

iterated integral

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