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 −0.23/12, 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.