Simpson's rule applied to sin(x), and also hardcoded

sin_simpson.py

#!/bin/python
# Approximates the definite integral of sin(x) using Simpson's rule.

from math import pi,sin,cos

def simpson_approximation(a, b, n):
	"Approximates the definite integral of sin(x) from a to b with respect to x with 'n' iterations"

	dx = (b-a)/n

	summation = 0

	for i in range(1, n):	# iterate from 1 to n-1
		if i % 2 == 1:		# if i is odd
			summation += 4 * sin(dx * i + a)
		elif i % 2 == 0:	# if i is even
			summation += 2 * sin(dx * i + a)

	summation += sin(a)
	summation += sin(b)

	approximate_integral = dx * summation / 3

	print("With n =",n,"the approximate integral of sin(x) from","%3f"%a,"to","%3f"%b,"with respect to x is", approximate_integral)

	return approximate_integral


a = 0
b = pi

n_list = [10, 100, 1000]

for n in n_list:
	approximation = simpson_approximation(a, b, n)
	print("This is a", "%3.2e" % (approximation/(cos(a) - cos(b)) - 1), "difference from the exact value of", -cos(a) + cos(b))
	print("")