Exercise 1.29. Simpson’s Rule is a more accurate method of numerical integration than the method illustrated above. Using Simpson’s Rule, the integral of a function f between a and b is approximated as
where h = (b - a)/n, for some even integer n, and yk = f(a + kh). (Increasing n increases the accuracy of the approximation.) Define a procedure that takes as arguments f, a, b, and n and returns the value of the integral, computed using Simpson’s Rule. Use your procedure to integrate cube between 0 and 1 (with n = 100 and n = 1000), and compare the results to those of the integral procedure shown above.
中间的那个列表可以拆成三个列表
\[ y_0+y_1+y_2+…+y_{n-1} +y_n \]
\[ y_1+y_2+y_3…+y_{n-2} +y_{n-1} \]
\[ 2y_1+2y_3+2y_5…+2y_{n-3} +2y_{n-1} \]
于是，我们可以写出程序
(define (integral f a b n)
(define h (/ (- b a) n))
(define (f-next x) (+ x h))
(define (f-next-2 x) (+ x h h))
(/ (* h
(+ (sum f (+ a h) f-next (- b h))
(sum f a f-next b)
(* 2 (sum f (+ a h) f-next-2 (- b h)))))
3.0))