From Riemanns sum to definite integral

Dear All!

I found this as an explanation!

Definite Integral Proof? - Yahoo! Answers

The problem is that I just dont understand it still form the first line:

F(X1) – F(a) = F’(C1) * (X1-a) = F’(C1)*ΔX1 = f(C1)* ΔX1(Punch)

Please, how do we get this?

Many thanks!!!

p.s. please, don't be angry I ask too much!!!

Re: From Riemanns sum to definite integral

Quote:

Originally Posted by

**Boo** Dear All!

I found this as an explanation!

Definite Integral Proof? - Yahoo! Answers
The problem is that I just dont understand it still form the first line:

F(X1) – F(a) = F’(C1) * (X1-a) = F’(C1)*ΔX1 = f(C1)* ΔX1(Punch)

Please, how do we get this?

Many thanks!!!

p.s. please, don't be angry I ask too much!!!

I do not think that anyone will be angry, certainly not me, at your question.

However, I will tell that I have given a semester long course on the theory of the integral in an attempt to completely answer your question.

Re: From Riemanns sum to definite integral

Can someone give me the hint?

Everyone I know avoids to answer....

What is $\displaystyle C_1$???

Re: From Riemanns sum to definite integral

Quote:

Originally Posted by

**Boo** Dear All!

I found this as an explanation!

Definite Integral Proof? - Yahoo! Answers
The problem is that I just dont understand it still form the first line:

F(X1) – F(a) = F’(C1) * (X1-a) = F’(C1)*ΔX1 = f(C1)* ΔX1(Punch)

C1 is a point **chosen** so that F(X1) – F(a) = F’(C1) * (X1-a). The fact that there **exist** such a value, C1, follows from the "mean value theorem". The fact that those equal F'(C1)*ΔX1 is just that ΔX1 is **defined** as (X1- a). And the fact that F'(C1)ΔX1= f(C1) follows from the fact "f" is **defined** as F'.

Please, how do we get this?

Many thanks!!!

p.s. please, don't be angry I ask too much!!![/QUOTE]