MCS 471 Computer Problem 4:
Integration using Maple Symbolically and Numerically

• Newton-Cotes Integration Rules
• Trapezoidal Rule
• Simpson's Rule
• Gaussian Quadrature

• Notes from Professor Hanson's Lectures
• Gerald and Wheatley, Chapter 4

Refer to the Class Maple Integration Help Page: Integrating a Function with Maple int

For more help, see also the Class Maple Pages

• Find integrals of particular test functions given below on their intervals (a,b) by the following three (3) methods

  A. Built-in Maple built-in integration function int,

  • Realizing you will need to use the Maple evalf function to get a floating point numerical answer to 10 digits;

  • Also use the Maple Plot function plot to plot the function to be integrated.

  B. Maple's "with(student);" trapezoid function with 60 subdivisions (61 points or nodes) each;

  C. Maple's "with(student);" simpson function with 60 subdivisions (61 points or nodes) each;

  D. Composite Gaussian Quadrature 2×GR₃,

  • i.e., an application GR₃ on each of two halves of the interval (a,b) using the weights and nodes,

  [w1, w2, w3]=[5./9.,  8./9.,   5./9.]    and    [t1, t2, t3]=[- sqrt(3./5.),   0.0,  sqrt(3./5.)],

  respectively, the nodes being standardized to the subinterval [- 1,1] by transforming x to a standard variable t for each half of [a,b] (i.e., on [a,(a+b)/2]+[(a+b)/2,b]; caution: note the division sign "/" in the weights and nodes is distinct from aone "1").

• Compute the per cent relative error of the 3 parts B - D relative to part A, for each of the following three (3) functions:

  1.   f1(x) = 29.7*x^8 - 23.2*x^7, on (2.25, 5.75).

     Remark: This is a test function that can easily be integrated exactly as a check to your methods.

     Question: For this first function, also compute the exact answer in your documentary comments. Does your Maple answer check with your exact calculus answer? Compute the per cent relative error from the exact answer for each method. Discuss the errors in your documentary comments.

  2.   f2(x) = {81.675/sqrt(10.89+x^2) if x > 0, else (7.5-5.6*x*exp(1.89*x))}, on (-5.25,+6.75).

     Remark 1: This is a nonsmooth example with piecewise definition, so what does that mean for the validity theoretical global error estimates? Would more points help?

     Remark 2: For Maple, use the Maple function "piecewise" given in the above Class Maple integration page to form the maple definition of the function, since if-then-else or procedure constructs lead to mysterious Maple errors
     (Do not trust any software: Check it out!).

  3.   f3(x) = 89.7*exp(-(x-1.853)^2)/sqrt(x^2+0.5e-12), on (0,4.75).

     Question: This is a nonsmooth nearly singular example, so what does that mean for the validity theoretical global error estimates? Would more points help? Give the Answer in your documentary comments.

Caution: There are three (4) methods for three (3) problems making twelve (12) items total.