Trapezoidal Rule

From a practical standpoint, it is desirable to have a good estimate of error. We shall first work an example and then state a theorem which gives an error estimate for the trapezoidal approximation.

Example 1

The trapezoidal approximation can be made as close to the definite integral as we want by taking Δx small. From a practical standpoint, however, it is helpful to know how small we should take Δx in order to be sure of a given degree of accuracy. For instance, suppose we need to know the definite integral to three decimal places. How small must we take Δx in our trapezoidal approximation? The answer is given by the Trapezoidal Rule, which gives an error estimate for the trapezoidal approximation.

The error in the trapezoidal approximation is the absolute value of the difference between the trapezoidal sum and the definite integral,

error =

An error estimate for the trapezoidal approximation is a function E(Δx), which is known to be greater than or equal to the error.

Thus if E(Δx) is an error estimate, the trapezoidal sum is within E(Δx) of the definite integral. If we want to be sure that the trapezoidal approximation is accurate to three decimal places - i.e., the error is less than 0.0005 - we choose Δx so that E(Δx) ≤ 0.0005. We are now ready to state the Trapezoidal Rule.

The proof is omitted.

Example1 (Concluded)

Example 2