Day 20 in MAT229

Today:

• Announcements:
  • Your tests are returned. Some comments:
    • Some of the problems were done on the board. Were you taking notes?
    • Some of the problems were worked out in your textbook. Did you do the assigned reading?
    • I told you, for example, that you'd have a problem dealing with the limit definition of the derivative. 15 of 24 of you actually could state the limit definition correctly (which means that 9 of 24 couldn't). It's the most important definition in calculus. Don't you think that you should know it well if you want to be considered a good calculus student?
    • You might like to consider making a "cheat sheet", which you continually update.

  • The key

• Today we'll finish section 7.7: numerical integration.
  • What's integration all about? Well, we usually start by going about calculating (signed) areas:
    

    "dA" is called an "infinitesimal" -- it's a tiny chunk of area -- tinier than anything you know ("vanishingly small")!

    What's numerical integration all about? We do pretty much the same thing, only we have

    

    where the are small, but not vanishingly small.

  • There are various schemes for numerical integration:
    • Left Rectangular
    • Right Rectangular
    • Midpoint
    • Trapezoidal
    • Simpson's rule.
    Let's see what unifies and what distinguishes them.

  • Three of the rules are so called "rectangle rules" (LRR, RRR, Midpoint Rule);
    

    but as the image to the right (above) and the graphical insight below show, we can think of the Midpoint rule as being a "Tangent rule":

    

    Be careful however not to confuse the midpoint and trapezoidal rule. The trapezoid represented above is not the same as the trapezoid of the trapezoidal rule.

    

  • The Trapezoidal rule is really just the average of the LRR and RRRs. This gives rise to an important observation, which I want to encourage you to think about, and here it is:
    When you have two estimates, you have a third.

    (their arithmetic average, in this case).

    

  • Simpson's rule is a "blend" (or weighted average) of the Midpoint and Trapezoidal rules, which perfectly balances the errors of the two to generate a better rule:
    

    Notice that the number of subintervals in Simpson's rule must be even.

    Now we can go further:

    When you have two estimates, you have infinitely many more.

    (their weighted arithmetic averages).

    

    Here are the error bounds, that illustrate that the errors of midpoint and trapezoidal are related, and suggest how to combine them to create a better method (Simpson's rule):

    

  • Here is a hand out which emphasizes this message. Let's take a look at this:
    • Mathematica code
    • Check midpoint calculation
    • Let's do the calculations by the formulas, for
      • Trapezoid
      • Simpson's
    • Check that we get the same things using the averages
    • Calculate the true value of the integral, and compare to the approximations. What's surprising?

  • Examples:
    • #29, p. 541
    • Others?

  • Links:
    • Here's a nice summary of the issue, focused on the use of Mathematica to illustrate them. Interesting even if you don't use Mathematica.
    • Numerical Integration on-line.