ex1_29.scm

;Exercise 1.29- Define a procedure that uses Simpson's Rule to compute the integral
; of a function. Also, compare this proc with another one in the text.

;to do this one, we need the cube procedure from the text:
(define (cube x)
  (* x x x))

;and also the sum procedure:
(define (sum term a next b)
  (if (> a b)
      0
      (+ (term a)
	 (sum term (next a) next b))))

;and we're going to compare the results to this integral proc:
(define (integral f a b dx)
  (define (add-dx x) (+ x dx))
  (* (sum f (+ a (/ dx 2.0)) add-dx b)
     dx))

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

;here's the exercise: define an integral-computing proc based on Simpson's rule:
(define (simpson f a b n)
  (define h (/ (- b a) n))
  (define (y k) (f (+ a (* k h))))
  (define (elem x)
    (cond ((or (= x 0) (= x n)) (y x))
	  ((odd? x) (* 4 (y x)))
	  ((even? x) (* 2 (y x)))))
  (* (/ h 3) (sum elem 0 inc n)))