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http://en.wikipedia.org/wiki/Mean_value_theorem#First_mean_value_theorem_for_integration"

Take a look at the Wikipedia proof. Now, wouldn't it be easier to prove it like this:

The ordinary mean value theorem says that

[tex]

G(b)-G(a)=(b-a)G'(\xi)

[/tex]

And the fundamental theorem of calculus says that

[tex]

G(b)-G(a)=\int_a^b G'(x)dx

[/tex]

So the conclusion is

[tex]

\int_a^b G'(x)dx=(b-a)G'(\xi)

[/tex]

Take a look at the Wikipedia proof. Now, wouldn't it be easier to prove it like this:

The ordinary mean value theorem says that

[tex]

G(b)-G(a)=(b-a)G'(\xi)

[/tex]

And the fundamental theorem of calculus says that

[tex]

G(b)-G(a)=\int_a^b G'(x)dx

[/tex]

So the conclusion is

[tex]

\int_a^b G'(x)dx=(b-a)G'(\xi)

[/tex]

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