i havent finished this and maybe i never will but i enjoyed what i read so far and need to ask: why do you think the trapezoidal approximation (the second thing you showed) doesn't improve much over the rectangle? initially, i thought "aha this is an example of taxicab metric behaviour" but it clearly isnt, we're going along slanted lines now, and anyway smaller rectangles is better (right?) so it cant be like taxicab at all...
There are probably more formal ways to argue this, but the idea I like is mentioned later in the post (both for Simpson's rule and for the Gauss rules), though not connected back to the trapeziums: odd-power terms don't contribute to an integral over an interval [-c, c]. We can always translate a sub-interval to be symmetric around x = 0, so a fitted linear term (being odd) between two points adds zero to the overall integral.
aha! so what you were losing on one side of the flat top of the rectangle you were gaining on the other side! (ie every gap under the curve induced by the flat surface corresponds to an extra bit outside the curve induced by the flat surface). nice.
i havent finished this and maybe i never will but i enjoyed what i read so far and need to ask: why do you think the trapezoidal approximation (the second thing you showed) doesn't improve much over the rectangle? initially, i thought "aha this is an example of taxicab metric behaviour" but it clearly isnt, we're going along slanted lines now, and anyway smaller rectangles is better (right?) so it cant be like taxicab at all...
There are probably more formal ways to argue this, but the idea I like is mentioned later in the post (both for Simpson's rule and for the Gauss rules), though not connected back to the trapeziums: odd-power terms don't contribute to an integral over an interval [-c, c]. We can always translate a sub-interval to be symmetric around x = 0, so a fitted linear term (being odd) between two points adds zero to the overall integral.
aha! so what you were losing on one side of the flat top of the rectangle you were gaining on the other side! (ie every gap under the curve induced by the flat surface corresponds to an extra bit outside the curve induced by the flat surface). nice.