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Assumptions
We will assume that f(x) is a scalar-valued function of a single variable x and, for error analysis, that f(x) has a continuous 2nd derivative f(2)(x) which we can compute.
Derivation
Given the function f(x) on [a, b], we can approximate this function on the interval by finding the linear polynomial which interpolates the two points (a, f(a)) and (b, f(b)). Using Lagrange interpolation, this interpolating line is
If we integrate this from a to b and simplify (left to the reader), we get the very simple formula
You will note that this is the area of a trapezoid of height b − a with sides of width f(a) and f(b).
Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.
