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].