Thread: 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

Please, how do we get this?
Many thanks!!!
p.s. please, don't be angry I ask too much!!!

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
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.

Can someone give me the hint?
Everyone I know avoids to answer....
What is ???

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

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]