Suppose that $f(x)$ is continuously differentiable on the interval [a,b].

• Let N be a positive integer
• Let $M = Max \{ |f'(x)| : a \leq x \leq b \}$
• Let $h = \frac{(b-a)}{N}$
• Let $R_N$ denote the "right endpoint"

Riemann Sum for the integral

$I = \int_a^b f(x) dx .$


In other words,

$R_N = \sum_{n=1}^N f(a + n h) h .$


Explain why the error, $E = | R_N - I |$, satisfies

$E \le \frac{M(b-a)^2}{N}.$


(I moved all the discussion that used to be here to the discussion tab. I still think it would be nice to put a polished proof of the estimate here. --Bell 14:49, 14 October 2008 (UTC))

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