Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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].

share|cite|improve this question
How do you define $\int_a^b f(x)dx$? – Chris Janjigian Sep 9 '12 at 13:24
What is $$\sum_{x=a}^b$$ anyway?? Are $\,a,b\,$ integers so that the variable $\,x\,$ can run between them? I think there's a serious mistake/typo in this question. – DonAntonio Sep 10 '12 at 3:01
That is the definition of the integral in most of the calculus books. Well, except for a few typos. – Keivan Sep 11 '12 at 3:04
Please see my question similar to yours.… – Mathlover Sep 11 '12 at 8:36
up vote 1 down vote accepted

If you mean you want a proof that shows that if



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


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

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

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.