Example 5.  How good are the Simpson's rule approximations to    that were calculated in Example 4?

Solution 5.

We can compare the error in these approximations.

The error for Simpson's 2D rule has the form      where h and k are the step sizes for the variables x and y, respectively.
For the above examples, we have the following results.

Remark.  Since both the step sizes were reduced by a factor of the remainder term   should be reduced by approximately .  

Therefore, the 2D Simpson's rule is behaving as predicted.

(c) John H. Mathews 2004