MCS 471 Computer Problem 3: Numerical Integration using Maple/Octave

• 4.4 Extrapolation Techniques
• 4.5 Newton-Cotes Integration Rules
• 4.6 Trapezoidal Rule
• 4.7 Simpson's Rule
• 4.9 Gaussian Quadrature

Refer to the Class Maple and/or Octave Pages, or in particular to the

Integrating a Function with Maple int

or

Octave Example for Evaluating an Integral.

See also the main class Maple and Octave pages for more help.  

Find integrals by the following three (3) methods

A.) Built-in Maple or Octave built-in integration function;
B.) Maple's "with(student);" trapezoid function with 20 subdivisions (21 points or nodes) each (Octave does not have built-in Trapezoidal Rule, so it has to be coded by hand);
C.) Maple's "with(student);" Simpson's function with 20 subdivisions (21 points or nodes) each; (Octave does not have built-in Simpson's Rule, so it has to be coded by hand);

1. f1(x) = 7*x^6 - 4*x^3 on (1,2).
   Remark 1: This is a test function that can easily be integrated exactly as a check to your methods.
   Remark 2: Give the exact answer in your documentary comments.
2. f2(x) = {2*sqrt(1+x^2) if x > 0, else (2+6*x^5)} on (-2.,+2.).
   Remark 1: This is a nonsmooth example with piecewise definition.
   Remark 2: For Maple, use the Maple function "piecewise" given in the above Class Maple integration page, since if-then-else or procedure constructs lead to mysterious Maple errors. Octave should be OK.
3. f3(x) = {exp(-x^4)/sqrt(x) if x > 0.25, else 13.767-47.1*x} on (0,5).
   Remark 1: This is a nonsmooth nearly singular example with piecewise definition.
   Remark 2: For Maple, use the Maple function "piecewise" given in the above Class Maple integration page, since if-then-else or procedure constructs lead to mysterious Maple errors. Also, Maple "piecewise" works better, if at all, with the nearly singular part in the "FunctionIfTrue" argument, rather than the "FunctionElse" argument. Octave should be OK. Caution: There are three (3) methods for three (3) problems making nine (9) items total.