I accept that the OP was not interested in a built-in command because (s)he wanted to produce a solution as an exercise. However I think it is reasonable to compare the OP's code with the Maple's built-in version, just for verification purposes
Looking at the codes supplied by OP and Kitonum (the two are equivalent), I was struck by thought that they might do interesting things depending on whether the number of points requested was 0 mod 3, 1 mod 3, or 2 mod 3.
The accuracy of the original version of what is now known as "Simpson's 3/8" rule, is completely independent of the number of points (although the name "3/8 rule" probably only makes real sense, when the number of points is a multiple of 3). The Maple help page ?Student[Calculus1][Simpson's 3/8 Rule] is actually quite informative in this respect
Consider the attached worksheet which will allow you to compare/contrast the result of OP's code, Kitonum's code, and the Maple's built-in version of "Simpson's 3/8" rule in two scenarios - where the number of points is small, and where the number of points is large
Notice how accurate the Maple built-in function is for small numbers of points, irrespective of whether the number of points is 0 mod 3, 1 mod 3, or 2 mod 3. OPcode (and Kitonum - because they produce identical results) are pretty close when the numbers of points is a multiple of 3 and pretty awful otherwise
Even with a large number of points, it is still noticeable that OPcode/Kitonum agrees much better with the Maple built-in command when the number of points is a multiple of 3.
Actually I expected complete agreement when the number of points is a multiple of 3, but I haven't got the time/energy to figure out why there is a discrepancy in this specific case - let's just say I'm backing the Maple version!
Why do I care??
Well I live quite close to where Thomas Simpson was born and is buried (and have visited the latter). A small plaque in a very old church isn't much reward for torturing generations of students, but hell, it's more than most of us will get, and he deserves some respect (even although many, much earlier, mathematicians had probably produced the same rules)