2D Trapezoidal and Simpson Rules

Background

    The trapezoidal rule and Simpson's rule for ordinary integrals can be extended to multiple integrals.

Theorem  (Trapezoidal 2D Rule)  Consider over the rectangle .  Given that the interval is subdivided into subintervals    of equal width    by using the equally spaced sample points    for  .   Also, assume that the interval is subdivided into subintervals    of equal width    by using the equally spaced sample points    for  .   
The composite Trapezoidal rule  is  

        
where

          

It can be shown that the error term is of the form  , that is

        .  

Remark.  The Trapezoidal rule had the pattern of weights    and the Trapezoidal 2D rule extends this pattern to a grid in the rectangle R.  
    
 

Theorem (Simpson's 2D Rule)   Consider over the rectangle .  Given that the interval is subdivided into subintervals    of equal width    by using the equally spaced sample points    for  .   Also, assume that the interval is subdivided into subintervals    of equal width    by using the equally spaced sample points    for  .   
The composite Simpson's rule  is  

        
where
          
    
It can be shown that the error term is of the form  , that is

        .  

Remark.  Simpson's rule had the pattern of weights    and Simpson's 2D rule extends this pattern to a grid in the rectangle R.  
    
 

Mathematica Subroutine (Trapezoidal 2D Rule). Object oriented programming.