Integral mean value theorem

[daudaudaudau]

http://en.wikipedia.org/wiki/Mean_value_theorem#First_mean_value_theorem_for_in tegration

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

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


And the fundamental theorem of calculus says that

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


So the conclusion is

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

[Office_Shredder]

You proved the case where \phi = 1 identically. Notice the wikipedia proof covers a far broader case than you did.

Also, you probably used the mean value theorem to prove the fundamental theorem of calculus, so this is circular

[daudaudaudau]

I see.