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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