Simpson's Rule

Main formula:

Simpson’s Rule (n must be even):

Hints and tips:

• Unlike the formulas for the Trapezoid, Midpoint, and Left and Right Endpoint Rules, this formula is too complicated to derive on the spot, so it is probably worth memorizing.

• However, to remember the pattern of the coefficients, it helps to remember that it comes from adding up overlapping sets of the form 1-4-1:

          1 - 4 - 1
                  1 - 4 - 1
                          1 - 4 - 1, etc.
  

• Simpson’s Rule is much more accurate than any of the Trapezoid, Midpoint, and Left and Right Endpoint Rules, especially for bigger values of n. However, it still takes about the same number of function evaluations.