### Trapezoidal Rule

The *trapezoidal rule* is defined by

Thus, the trapezoidal rule is driven by the

*average*of the derivative estimates at times and . The method is implicit in either forward or reverse time.

The trapezoidal rule gets its name from the fact that it approximates an integral by summing the areas of trapezoids. This can be seen by writing Eq.(7.12) as

An interesting fact about the trapezoidal rule is that it is
equivalent to the *bilinear transform* in the linear,
time-invariant case. Carrying Eq.(7.12) to the frequency domain
gives

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