The error for the trapezoidal rule may be found by integrating the corresponding Taylor series and applying integration by parts. Taylor's series gives us that:

where ξ is in the interval [*a*, *x*].

Integrating each side with respect to the variable *x*
from *a* to *b*, we get the expression:

Recall from calculus the technique of integration by parts. In this
case, may rewrite the integral of *x* f^{(1)}(*x*) as:

Substituting this into our expression, we get:

Exchanging the integral and the left-hand side, dividing each side by −2, and collecting appropriately, we get:

Note, we sort of cheated here, because ξ depends on *x*, however
we may replace f^{(2)}(ξ) by the average value:

# Example of Error

Consider the integral of cos(*x*) from 0.2 to 0.4. The correct
value of this integral is 0.1907490115, 0.5⋅(cos(0.2) + cos(0.4))⋅0.2 = 0.1901127572,
and the difference between these is −0.0006362543. If we calculate the
average value of the 2nd derivative and multiply this by −0.2^{3}/12, we
get −0.0006358300384, and thus our approximation of the error is very
close (less than 0.1% error in our estimation of the error).

Copyright ©2005 by Douglas Wilhelm Harder. All rights reserved.