# Math Help - Riemann sum

1. ## Riemann sum

Use the Archimedes-Riemann Theorem to show that for $0 \leq a < b$, we have $\int ^b_a xdx = \frac {b^2-a^2}{2}$

I'm trying to find my sum but I can't seem to remember how to do it, any hints ,please?

2. $
\mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {\left\{ {\frac{{\left( {b - a} \right)k}}
{n}} \right\}} \cdot \frac{{b - a}}
{n} = \int_a^b x dx
$

3. Yes, thank you.

But now I'm still stuck... How would I go on with this problem?

4. Originally Posted by Nacho
$
\mathop {\lim }\limits_{n \to \infty } \sum\limits_{k = 1}^n {\left\{ {\frac{{\left( {b - a} \right)k}}
{n}} \right\}} \cdot \frac{{b - a}}
{n} = \int_a^b x dx
$

\begin{aligned}\lim_{n\to\infty}\sum_{k=1}^{n}\lef t\{{\color{red}a}+\frac{(b-a)k}{n}\right\}\cdot\frac{b-a}{n}&=\lim_{n\to\infty}\sum_{k=1}^{\infty}\left\{ \frac{a(b-a)}{n}+\frac{(b-a)^2k}{n^2}\right\}\\
${\color{red}\star}~\sum_{k=1}^{n}k=\frac{n(n+1)}{2 }~~\sum_{k=1}^{n}c=cn$