# The basic estimate for the rectangle method

Suppose that $f(x)$ is a continuously differentiable function on $[a,b]$. Let $N$ be a positive integer and let $M=\text{Max}\ \{ |f'(x)|: a\le x\le b\}$. Define $R_N$ to be the the right endpoint Riemann Sum

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

where $\Delta x = (b-a)/N$, and let

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

We shall prove that the error, $E=|R_N-I|$ satisfies the estimate,

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

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