# Thread: Definition of integral

1. ## Definition of integral

This is the definition of an integral, and I understand it, but how do you read it out loud, and what does the $\displaystyle dx$ mean?

$\displaystyle \int_a^b f(x)dx = \lim\limits_{n \to \infty} \sum\limits^n_{i=1}f(x_i^*)\Delta x$

2. It is "integral from a to b of f(x) with respect to x". Or you can say "It's the integral of f with respect to x and goes from a to b". dx means with respect to.

3. 'kay thanx. What about the right side? I would assume that reads "The limit of the sum F of x sub i times delta x as i approaches n from 1 as n approaches infinity." Am I right?

4. Not quite. You would say "the limit as n approaches infinity of the sum from i is equal to 1 to n of F of x i star times delta x". That's pretty nasty to just say it straight away.

You can make it easier for yourself by saying something like "the limit of the sum as n approaches infinity. The sum is from i equal to 1 to n of F of x i star times delta x "

5. OK, so the dx means with respect to x, but what is that used for? On my calculator it makes me type in the variable too, but shouldn't it be obvious which variable to use when it's the only variable in the equation?

6. It should totally be obvious, yes. In fact some theorems in analysis that deal with integrals of one variable sometimes don't even write the dx. However, it's good practice. Not to mention that it's difficult for calculators to "know" which variable you are integrating with respect to when you have arbitrary constants lurking in your expression.

However, if you look at this tool here integral 1&#47;sqrt&#40;1&#43;x&#94;2&#41; - Wolfram|Alpha, I have inputted an integrand without the dx and the program recognizes it.

However, when you put in two variables like in this example integral 1&#47;sqrt&#40;x&#94;2&#43;y&#94;2&#41; - Wolfram|Alpha, it will just pick one of the variables and integrate with respect to it.