# L-9 MCS 507 Mon 12 Sep 2011 : trapezoidal rule

# The trapezoidal rule approximates the
# integral of a function f(x) over [a,b]
# via (f(a) + f(b))*(b-a)/2.
# With one extra function evaluation we
# return an error estimate.

def traprule(f,a,b):
   "trapezoidal rule for f(x) over [a,b]"
   T = (b-a)*(f(a) + f(b))/2
   m = (a+b)/2
   R = T/2 + (b-a)*f(m)/2
   return R, abs(T-R)

import math
s = 'integrating exp() over '
print s + '[a,b]'
a = input('give a : ')
b = input('give b : ')
y, e = traprule(math.exp,a,b)
print s + '[%.1E,%.1E] : ' % (a,b)
print 'the approximation : %.15E' % y
print 'an error estimate : %.4E' % e
E = math.exp(b) - math.exp(a)
print '  the exact value : %.15E' % E