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How to prove that $\int^a_bf(x)dx=\lim_{\Delta x\rightarrow0}\sum_{x=a}^bf(x)\Delta x$ (which $\int^a_bf(x)dx=g(a)-g(b)$ where $\frac{d}{dx}g(x)=f(x))$?

I know this is a very basic thing of integration but it seems that I can't its proof anywhere. Please help me... Thank you.


Maybe I should say it in this way: prove that $\int^a_bf(x)dx$ is the area between the curve y=f(x) and x-axis in the interval [b,a].

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    Please see my question similar to yours. http://math.stackexchange.com/questions/123048/the-net-signed-area-between-t-0-y-0-t-x-and-y-ft2012-09-11

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If you mean you want a proof that shows that if

$F'(x)=f(x)$

then

$F(b)-F(a)=\lim_{n \to \infty} \Delta x\sum_{x=0}^{n-1} f(x+n\Delta x)$

where

$\Delta x=\frac{b-a}{n}$

I suggest you look in to the fundamental theorem of calculus.