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

• 4.2 Derivatives from Difference Tables
• 4.3 Higher-Order Dervatives
• 4.4 Extrapolation Techniques
• 4.5 Newton-Cotes Integration Rules
• 4.6 Trapezoidal Rule
• 4.7 Simpson's Rule
• 4.9 Gaussian Quadrature
• 4.10 Adaptive Integration

Refer to the Class Octave and Maple Pages, or in particular to the "Integrating a Function with Maple int (newly Revised!):"

http://www.math.uic.edu/~hanson/MAPLE/MapleIntegral.html

or "Octave Example for Evaluating an Integral:"

http://www.math.uic.edu/~hanson/Octave/OctaveIntegralEG.html

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

Find the integral of the following three functions:

1. f1(x) = x**5 - x**3 on (0,2).
   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 you documentary comments.
2. f2(x) = {sqrt(x) if x > 0, else x**4} on (-1.,+1.).
   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^3)/sqrt(x) if x > 0.1, else 10.-68.4*x} on (0,4).
   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.