second integral mean-value theorem

second integral mean-value theorem

If the real functions

f

and

g

are

continuous

and

f

monotonic

on the

interval

[

a

,

b

]

,  then the equation

∫

a

b

f

⁢

(

x

)

⁢

g

⁢

(

x

)

⁢

𝑑

x

=

f

⁢

(

a

)

⁢

∫

a

ξ

g

⁢

(

x

)

⁢

𝑑

x

+

f

⁢

(

b

)

⁢

∫

ξ

b

g

⁢

(

x

)

⁢

𝑑

x

(1)

is true for a value

ξ

in this interval.

Proof.

We can suppose that

f

⁢

(

a

)

≠

f

⁢

(

b

)

since otherwise any value of

ξ

between

a

and

b

would do.

Let’s first prove the auxiliary result, that if a

function

φ

is continuous on an open interval

I

containing

[

a

,

b

]

then

lim

h

→

0

⁡

∫

a

b

φ

⁢

(

x

+

h

)

-

φ

⁢

(

x

)

h

⁢

𝑑

x

=

φ

⁢

(

b

)

-

φ

⁢

(

a

)

.

(2)

In fact, when we take an

antiderivative

Φ

of

φ

, then for every nonzero

h

the function

x

↦

Φ

⁢

(

x

+

h

)

-

Φ

⁢

(

x

)

h

is an antiderivative of the integrand of (2) on the interval

[

a

,

b

]

.  Thus we have

∫

a

b

φ

⁢

(

x

+

h

)

-

φ

⁢

(

x

)

h

⁢

𝑑

x

=

Φ

⁢

(

b

+

h

)

-

Φ

⁢

(

x

)

h

-

Φ

⁢

(

a

+

h

)

-

Φ

⁢

(

x

)

h

→

φ

⁢

(

b

)

-

φ

⁢

(

a

)

as

⁢

h

→

0

.

The given functions

f

and

g

can be extended on an open interval

I

containing

[

a

,

b

]

such that they remain continuous and

f

monotonic.  We take an antiderivative

G

of

g

and a nonzero number

h

having small

absolute value

.  Then we can write the

identity

∫

a

b

f

⁢

(

x

+

h

)

⁢

G

⁢

(

x

+

h

)

-

f

⁢

(

x

)

⁢

G

⁢

(

x

)

h

⁢

𝑑

x

=

∫

a

b

f

⁢

(

x

+

h

)

⁢

[

G

⁢

(

x

+

h

)

-

G

⁢

(

x

)

]

h

⁢

𝑑

x

-

∫

a

b

f

⁢

(

x

+

h

)

-

f

⁢

(

x

)

h

⁢

G

⁢

(

x

)

⁢

𝑑

x

.

(3)

By (2), the left hand side of (3) may be written

∫

a

b

f

⁢

(

x

+

h

)

⁢

G

⁢

(

x

+

h

)

-

f

⁢

(

x

)

⁢

G

⁢

(

x

)

h

⁢

𝑑

x

=

f

⁢

(

b

)

⁢

G

⁢

(

b

)

-

f

⁢

(

a

)

⁢

G

⁢

(

a

)

+

ε

1

⁢

(

h

)

(4)

where

ε

1

⁢

(

h

)

→

0

as

h

→

0

.  Further, the function

x

↦

{

f

⁢

(

x

+

h

)

⁢

[

G

⁢

(

x

+

h

)

-

G

⁢

(

x

)

]

h

for

h

≠

0

f

⁢

(

x

)

⁢

g

⁢

(

x

)

for

h

=

0

is continuous in a rectangle

a

≤

x

≤

b

,

-

δ

≤

h

≤

δ

,  whence we have

∫

a

b

f

⁢

(

x

+

h

)

⁢

[

G

⁢

(

x

+

h

)

-

G

⁢

(

x

)

]

h

⁢

𝑑

x

=

∫

a

b

f

⁢

(

x

)

⁢

g

⁢

(

x

)

⁢

𝑑

x

+

ε

2

⁢

(

h

)

(5)

where

ε

2

⁢

(

h

)

→

0

as

h

→

0

.  Because of the monotonicity of

f

, the expression

f

⁢

(

x

+

h

)

-

f

⁢

(

x

)

h

does not change its sign when

a

≤

x

≤

b

.  Then the usual

integral mean value theorem

guarantees for every

h

(sufficiently near 0) a number

ξ

h

of the interval

[

a

,

b

]

such that

∫

a

b

f

⁢

(

x

+

h

)

-

f

⁢

(

x

)

h

⁢

G

⁢

(

x

)

⁢

𝑑

x

=

G

⁢

(

ξ

h

)

⁢

∫

a

b

f

⁢

(

x

+

h

)

-

f

⁢

(

x

)

h

⁢

𝑑

x

,

and the auxiliary result (2) allows to write this as

∫

a

b

f

⁢

(

x

+

h

)

-

f

⁢

(

x

)

h

⁢

G

⁢

(

x

)

⁢

𝑑

x

=

G

⁢

(

ξ

h

)

⁢

[

f

⁢

(

b

)

-

f

⁢

(

a

)

+

ε

3

⁢

(

h

)

]

(6)

with

ε

3

⁢

(

h

)

→

0

as

h

→

0

.  Now the equations (4), (5) and (6) imply

f

⁢

(

b

)

⁢

G

⁢

(

b

)

-

f

⁢

(

a

)

⁢

G

⁢

(

a

)

+

ε

1

⁢

(

h

)

=

∫

a

b

f

⁢

(

x

)

⁢

g

⁢

(

x

)

⁢

𝑑

x

+

ε

2

⁢

(

h

)

+

G

⁢

(

ξ

h

)

⁢

[

f

⁢

(

b

)

-

f

⁢

(

a

)

+

ε

3

⁢

(

h

)

]

.

(7)

Because

f

⁢

(

b

)

-

f

⁢

(

a

)

≠

0

,  the expression

G

⁢

(

ξ

h

)

has a limit

L

for

h

→

0

.  By the

continuity

of

G

there must be a number

ξ

between

a

and

b

such that

G

⁢

(

ξ

)

=

L

.  Letting then

h

tend to 0 we thus get the limiting equation

f

⁢

(

b

)

⁢

G

⁢

(

b

)

-

f

⁢

(

a

)

⁢

G

⁢

(

a

)

=

∫

a

b

f

⁢

(

x

)

⁢

g

⁢

(

x

)

⁢

𝑑

x

+

G

⁢

(

ξ

)

⁢

[

f

⁢

(

b

)

-

f

⁢

(

a

)

]

,

which finally gives

∫

a

b

f

⁢

(

x

)

⁢

g

⁢

(

x

)

⁢

𝑑

x

=

f

⁢

(

a

)

⁢

[

G

⁢

(

ξ

)

-

G

⁢

(

a

)

]

+

f

⁢

(

b

)

⁢

[

G

⁢

(

b

)

-

G

⁢

(

ξ

)

]

=

f

⁢

(

a

)

⁢

∫

a

ξ

g

⁢

(

x

)

⁢

𝑑

x

+

f

⁢

(

b

)

⁢

∫

ξ

b

g

⁢

(

x

)

⁢

𝑑

x

Q.E.D.

Title

second integral mean-value theorem

Canonical name

SecondIntegralMeanvalueTheorem

Date of creation

2013-03-22 18:20:21

Last modified on

2013-03-22 18:20:21

Owner

pahio (2872)

Last modified by

pahio (2872)

Numerical id

12

Author

pahio (2872)

Entry type

Theorem

Classification

msc 26A48

Classification

msc 26A42

Classification

msc 26A06

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