I know the general method to derive the trapezoid rule is with Taylor series, or, you know, to just look at the trapezoids and figure out the rule. However, I feel that for such a simple rule, there must exist some other, perhaps simpler, derivations. I can't seem to find any online, however. Are there indeed more ways to derive the trapezoid rule? By simpler, I mean a simpler algebraic method.
Derivations of the trapezoid rule
alternative-proofintuitionnumerical methods
Best Answer
Adding up the areas of the trapezia is very simple and natural... How much simpler could it be? You can also think of this rule (or any other numerical quadrature) as substituting $f$ by a constant function. The value of that constant function is a weighed average of values of $f$ (nodes with one neighbor have weight $\frac 1n$ and nodes with a single neighbor have weight $\frac{1}{2n}$).