"""
Numerical integration or quadrature for a smooth function f with known values at x_i

This method is the classical approch of suming 'Equally Spaced Abscissas'

method 2:
"Simpson Rule"

"""
def method_2(boundary, steps):
# "Simpson Rule"
# int(f) = delta_x/2 * (b-a)/3*(f1 + 4f2 + 2f_3 + ... + fn)
    h = (boundary[1] - boundary[0]) / steps
    a = boundary[0]
    b = boundary[1]
    x_i = make_points(a,b,h)
    y = 0.0
    y += (h/3.0)*f(a)
    cnt = 2
    for i in x_i:
        y += (h/3)*(4-2*(cnt%2))*f(i)
        cnt += 1
    y += (h/3.0)*f(b)
    return y

def make_points(a,b,h):
    x = a + h
    while x < (b-h):
        yield x
        x = x + h

def f(x): #enter your function here
    y = (x-0)*(x-0)
    return y

def main():
    a = 0.0 #Lower bound of integration
    b = 1.0    #Upper bound of integration
    steps = 10.0        #define number of steps or resolution
    boundary = [a, b]    #define boundary of integration
    y = method_2(boundary, steps)
    print('y = {0}'.format(y))

if __name__ == '__main__':
        main()