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.
The sum procedure above generates a linear recursion. The procedure can be rewritten so that the sum is performed iteratively.
Show how to do this by filling in the missing expressions in the
(define (sum term a next b)
(define (iter a result)
(iter <??> <??>)))
(iter <??> <??>))
The sum procedure is only the simplest of a vast number of similar abstractions that can be captured as higher-order procedures. Write an analogous procedure called product that returns the product of the values of a
function at points over a given range.
Show how to define factorial in terms of
product. Also use product to compute approximations to
π using the formula 
If your product procedure generates a recursive process, write one that generates an iterative process. If it generates an iterative process, write one that generates a recursive process.
 The intent of exercises 1.31 - 1.33 is to demonstrate the expressive power that is attained by using an appropriate abstraction to consolidate many seemingly disparate operations. However, though accumulation and filtering are elegant ideas, our hands are somewhat tied in using them at this point since we do not yet have data structures to provide suitable means of combination for these abstractions. We will return to these ideas in section 2.2.3 when we show how to use sequences as interfaces for combining filters and accumulators to build even more powerful abstractions. We will see there how these methods really come into their own as a powerful and elegant approach to designing programs. [back]
 This formula was discovered by the seventeenth-century English mathematician John Wallis. [back]
Show that sum and product
(exercise 1.31) are both special cases of a still more
general notion called accumulate that combines a collection of terms, using some general accumulation function:
(accumulate combiner null-value term a next b)
Accumulate takes as arguments the same term and range
specifications as sum and product, together with a combiner procedure (of two arguments) that specifies how the current
term is to be combined with the accumulation of the preceding terms
and a null-value that specifies what base value to use
when the terms run out. Write accumulate
and show how sum and product can both
be defined as simple calls to accumulate.
If your accumulate procedure generates a recursive process, write one that generates an iterative process. If it generates an iterative process, write one that generates a recursive process.
You can obtain an even more general version of accumulate
(exercise 1.32) by introducing the notion of a filter on the terms to be combined. That is, combine only those
terms derived from values in the range that satisfy a specified condition. The resulting filtered-accumulate abstraction takes
the same arguments as accumulate, together with an additional
predicate of one argument that specifies the filter. Write filtered-accumulate as a procedure. Show how to express the
following using filtered-accumulate:
the sum of the squares of the prime numbers in the interval a to
b (assuming that you have a prime? predicate already written)
the product of all the positive integers less than n
that are relatively prime to n (i.e., all positive integers i < n such that GCD(i, n) = 1).