Day 22 in Mat229

Today:

• Announcements:
  • Our quiz today is over integration techniques (non-numerical!).
  • Today we'll start to wrap up numerical integration. The lab tomorrow will cover writing methods for doing these quickly and easily in Mathematica.

• Last time:
  • We discussed error bounds for three methods. The trapezoidal type methods (trap and midpoint) require knowing how to bound the second derivative; Simpson's requires bounding the fourth.

• Today:
  • We'll continue exploring the error bounds.

  • I'll have you repeat our work on a simple single panel, but with a different integral: let's approximate the integral \[ \int_0^1 x^3 d x \] using a step-size of 1.

  • Here are the "error bounds" for the top three methods, showing that the errors of midpoint and trapezoidal are related, and suggest how to combine them to create a better method (Simpson's rule):
    

  • To do, and questions:
    • compare the results using our five methods
    • How do we know which will be over-estimates and which under-estimates?
    • compare the absolute errors in our five methods
    • compare the error to our error bounds in our three top methods: what are your k2 and k4, bounding derivatives?
    • Are we surprised by anything?

  • Now let's head back to our Numerical Integration handout to consider a few more issues there (Numerical Integration (nb)). In particular,
    • let's look at the calculation of the number of subintervals to use to achieve a given error (top of page 4). What are the steps?
      • The method is trapezoidal: what is its error formula?
      • What derivative do we need to consider? Can you compute it? Can you bound it on the interval of interest?
      • Solve for \(n\): make sure that you provide an integral value.
      • Let's have a look.....

    • Let's look at that last problem on page 5.

  • Then we'll have a quiz!

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