I am well aware of five approximate integration techniques:
1) Left endpoint approximation
2) Right endpoint approximation
3) Midpoint rule
4) Trapezoidal Rule
5) Simpson's Rule
It is very natural to dervive all of these rules using Taylor Series (all you have to do is choose your centre point , integrate term by term, shave of a few terms, and use approximate differentiaton place of the derivatives to obtain them!)
However, no matter what I do, I can't seem to obtain the trapezoidal rule. Just wondering if anyone has any ideas!