  Using elementary methods, we define and derive a particular weighted average
of the trapezoidal and composite trapezoidal rules and show that this
approximation, as well as its composite, is straightforward in computation.
This approximation and its composite, in their general forms, are shown to have
predictable error patterns; thus, an extrapolation method can be used to
increase the accuracy. We then derive the necessary weights to use an
extrapolation method to reduce error and converge more quickly than Romberg
integration by allowing for improved accuracy with fewer necessary
subintervals. The procedure necessary to implement this alternative method is
then carefully described, followed by two examples.
