(define (sum term a next b)
  (if (> a b)
      0
      (+ (term a)
         (sum term (next a) next b))))

(define (inc x) (+ x 1))

(define (simpsons-rule f a b n)
  (define h (/ (- b a) n))
  (define (term k)
    (* (f (+ a (* k h)))
       (cond ((= k 0) 1)
             ((= k n) 1)
             ((even? k) 2)
             (else 4))))
  (/ (* h
        (sum term 0 inc n))
     3))

(define (cube x) (* x x x))

; Simpson's rule is much more accurate than the naive integration method

; 1 ]=> (simpsons-rule cube 0.0 1.0 100)
; 
; ;Value: .24999999999999992
; 
; 1 ]=> (simpsons-rule cube 0.0 1.0 1000)
; 
; ;Value: .2500000000000003
; 
; 1 ]=> (simpsons-rule cube 0.0 1.0 10)
; 
; ;Value: .25000000000000006