Newton-Cotes Integration Formulas - 1. The Trapezoid Rules

The Newton-Cotes integration formulas are common approximations in many numerical integration methods. Simply, the complicated function would be replaced with an approximation function.

f(x) and f1(x) are the complicated function and an approximation function, repectively.

Using easy approximation function won't bring a perfect answer like an analytical way, called for finite integration. However, it would be a best way when the function is too complicated to be calculated with the techniques we learned in calculus classes.

There are several types of approximations so we are going to start with trapezoid rule.

1. Trapezoid rule

   This equations is derived from the following interpolation equation that will be posed soon.

   After rearrangement f1(x), it should be integrated from a to b.

 Finally we could get the first equation.

  Example question.
     

from a=0 to b=0.8. 

  Solve
        1) Calculate the equation analytically.

      
             = 1.640533

         2) Use the Trapezoid equation

              f(0) = 0.2, f(0.8) = 0.232
        

          3) Calculate Error

              [(Analytical value - approximation value) / Analytical Value] x 100 = ?
              [(1.640533 - 0.1728) / 1.640533] x 100 = 89.5 %.

           4) Interpretation
              The trapezoid rule is based on interpolation so lower number of segments will cause higher error percentages. The following two graphic can show the rule easily.                
 
             
                                one segment                                       two segments