Example 7.  Use the composite Simpson's rule for multiple integrals to numerically approximate the iterated integral  .

Solution 7.

We use  n = 20 and  m = 5 in our computations.
The integrand is:

The curves bounding the region are:

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

Before we carry out the quadrature, we must fix "m" the number of vertical subdivisions to be used along each of the vertical segments between the curves y = c[x] and y = d[x].  

The variable "m" is global and is used in the numerical quadrature subroutine to define the function  .  

Now, fix  m  and perform numerical multiple integration.

How good was numerical quadrature ?  
Usually the special functions involved in the analytic solution of    are not usually covered in the standard calculus sequence.  
For your information,. the solution using Mathematica is found as follows:

Integrate    with respect to the variable y.  

Compute the definite integral  .

Integrate    with respect to the variable x.  

Compute the definite integral  

We have found the value of the iterated integral

(c) John H. Mathews 2004