Midpoint Rule

Theorem  (Midpoint Rule)  Consider over , where . The midpoint rule is  

    .  

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

    .  

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

    .
 

Composite Midpoint Rule

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

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

    .  

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

    .  

Remainder term for the Composite Midpoit Rule

Corollary  (Midpoint Rule: Remainder term)  Suppose that is subdivided into  m  subintervals    of width  .   The composite midpoint 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 Midpoint Rule.  To approximate the integral  

    ,  

by sampling at the equally spaced points    for  ,  where  .  

Mathematica Subroutine (Midpoint Rule).

Or you can use the traditional program.

Mathematica Subroutine (Midpoint Rule).

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