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 simpson 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.
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(define (cube x) (* x x x)) (check-equal? (round (* 100 (simpson cube 0 1 100))) 25.0) (check-equal? (round (* 100 (simpson cube 0 1 1000))) 25.0) (check-equal? (floor (* 1000 (simpson cube 0 1 100))) 249.0) (check-equal? (floor (* 1000 (simpson cube 0 1 1000))) 250.0)