Example 4.  Use the composite Simpson's rule for multiple integrals to numerically approximate the iterated integral  .  
Remark. This is the volume of the solid bounded by the surface  ,  that lies above the square    in the xy-plane.

Solution 4.

For illustration, we use the grid with  m = 5 and  n = 5.
Enter the integrand.

The region of integration in the xy-plane can be seen in the following graphical plot.

Execute our subroutine Simpson2D.

If you need more decimal places, Mathematica can get them.

Next, we find the approximation by using a finer mesh grid.

Execute our subroutine Simpson2D.

If you need more decimal places, Mathematica can get them.

(c) John H. Mathews 2004