March 31 2008

MAIN POINTS
The text describes how we can estimate the definite integral of a function through "numerical quadrature," which is basically performed by selecting distinct points within the desired interval, evaluating them at the function, finding the Lagrange Interpolating Polynomial of these points, and then integrating the resulting polynomial. The error term for this integral looks nearly impossible to estimate, because it is inside both a product notation, and an integral.
The text describes the trapezoidal rule next, which is taught in high school, and just creates neighboring trapezoids out of function values. Simpson's rule finds the second-degree Lagrange polynomial between two function values. Its error term includes a fourth derivative, so knowing the fourth-derivative behavior of the function can give an estimate of how exact the answer will be.
Finally, the text makes a generalization of this class of Newton-Cotes' methods. I don't really understand how this generalization works, and how the n=... values of these methods are built up.

CHALLENGES
As written above, how are Newton-Cotes formulas built up? What's the difference between open and closed? Is it basically the Lagrange interpolating polynomial of n points being integrated? Is the basic integration model with the rectangles inside the function the model for n=0?

REFLECTION
This is more interesting than the normal approximation we learned in HS calc. As n increases, the amount of calculation seems to increase. The error term may or may not drop, it seems to depend on h and the upper derivative behavior of f(x).