Trapezoidal Rule for Numerical Integration

Theorem  (Trapezoidal Rule)  Consider over , where . The trapezoidal rule is  

    .  

This is an numerical approximation to the integral of over and we have the expression  

    .  

The remainder term for the trapezoidal rule is  ,  where lies somewhere between , and have the equality  

    .

 

Composite Trapezoidal Rule

    An intuitive method of finding the area under a curve y = f(x)  is by approximating that area with a series of trapezoids that lie above the intervals  .  When several trapezoids are used, we call it the composite trapezoidal rule.  

Theorem  (Composite Trapezoidal Rule)  Consider over .  Suppose that the interval is subdivided into  m  subintervals    of equal width    by using the equally spaced nodes    for  .   The composite trapezoidal rule for m subintervals is  

    .  

This is an numerical approximation to the integral of over and we write  

    .  

Remainder term for the Composite Trapezoidal Rule

Corollary  (Trapezoidal Rule: Remainder term)  Suppose that is subdivided into  m  subintervals    of width  .   The composite trapezoidal rule  

      

is an numerical approximation to the integral, and  

    .  

Furthermore, if ,  then there exists a value  c  with  a < c < b  so that the error term    has the form

    .  

This is expressed using the "big " notation  .  

Remark.  When the step size is reduced by a factor of the error term   should be reduced by approximately .  

Algorithm Composite Trapezoidal Rule.  To approximate the integral  

    ,  

by sampling at the equally spaced points    for  ,  where  .  Notice that    and  .  

Mathematica Subroutine (Trapezoidal Rule).

Or you can use the traditional program.

Mathematica Subroutine (Trapezoidal Rule).

Example  Numerically approximate the integral    by using the trapezoidal rule with  m = 1, 2, 4, 8, and 16  subintervals.

Recursive Integration Rules

Theorem (Successive Trapezoidal Rules)  Suppose that    and the points    subdivide into    subintervals equal width  .  The trapezoidal rules obey the relationship  

    .  

Definition (Sequence of Trapezoidal Rules)  Define  ,  which is the trapezoidal rule with step size  .  Then for each    define    is the trapezoidal rule with step size  .  

Corollary (Recursive Trapezoidal Rule)  Start with  .  Then a sequence of trapezoidal rules    is generated by the recursive formula  

      for  .  

where  .  

The recursive trapezoidal rule is used for the Romberg integration algorithm.