Lab for Numerical Multiple Integrals

Module for Simpson's Rule for 2D

Algorithm. Composite Simpson Rule for 2D.

To numerically approximate the integral .

First, apply Simpson's rule using m subintervals of to f[x,y]
and define the function F[x]. Then, apply Simpson's rule using n subintervals of to F[x].
Remark. To make F[x] "look like a function of x" we fix the number of vertical subdivisions "m" as a global variable.
Use the following two Mathematica subroutines.

First load Mathematica's graphics package "FilledPlot".

Report to be handed in.

Computer Exercises



 




 

Exercise 1. Use the composite Simpson's rule for multiple integrals to
numerically approximate the iterated integral .


Use n = 8 and m = 8 in your computations.
The integrand is:

The curves bounding the region are lines:

The region of integration 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:

Exercise 2. Use the composite Simpson's rule for multiple integrals to
numerically approximate the iterated integral .


Use n = 20 and m = 5 in your computations.
The integrand is:

The curves bounding the region are:

The region of integration 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:

(c) John H. Mathews, 1998