Project Synergy 

Assignment using web object 

Author: Don Robinson (Allegany College of Maryland)

Web object “Numerical Integration Rules” by Dan Sloughter at Furman University Numerical Integration

 Background: A good assignment for Calculus I class, almost a stand alone on simple numerical integration techniques. If this was assigned from a class, not much lecture time would be required on the subject, unless you wanted to prove the conjecture. (Part II number 5)

Students should know how to integrate  by trig substitution, but if not we can still use the web object to demonstrate accuracy of numerical techniques as well as increased accuracy due to smaller choice of delta. The true answer for this question is arctan(4) + arctan(2) = 2.4329663814621229681 (20 significant digits) so comparison to the actual answer can be made.

 

I. Part 1.

  1. Visit the web object listed above and find the default approximate answer, and compare with the answer above. It is remarkable close, but really should not be. What was your answer? Why is it close? Why was that ‘lucky’?
  2. Double the number of subintervals and try again. Describe what happens to the answer. Was it unexpected? Describe why this result occurred.
  3. Dou you think the midpoint rule would be more accurate? Try with default 4 subintervals and see, then double to 8 subintervals, then 16 and note the result.
  4. Now continue with the midpoint rule, doubling until you have accuracy to the nearest 10 thousandth. How many subintervals n were needed?
  5.  Repeat 4 using the trapezoid rule. What is your value of n?
  6.  Repeat 4 using Simpson’s rule. What is the value of n?
  7. If you were an engineer and needed an answer to such a question, which method would you pick?

II. Part 2.

  1. Compute the value (let’s call it M) using the midpoint rule and n =16 subintervals, and compute the value (let’s call it T) using the trapezoid rule with n =16 subintervals.
  2. Now compute the ‘weighted average’ . Is this a good approximation?
  3. Compute the value using Simpson’s rule with n = 16 ‘rectangles’.
  4. Compare your answers to 2 and 3 and make a conjecture.
  5. (The hard part!!! Experts tread here only!) Prove your conjecture.