Alternative to the Romberg Method of Estimating the Definite Integral
  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. 
Michael Brandon Youngberg
math.NA