Integral approximation using the trapezoidal, midpoint, and simpson approximation technique.

integral.py

# the midpoint approximation of the integral of x*x from [0 1], with 10 subdivisions
# integral.midpoint(lambda x: x*x,0,1,10)

def trapezoidal(f, start, end, n, _area=0, _n=0):
    if _n is 0:
        return trapezoidal(f, start, end, n, _area, n)    
    elif n is 0:
        return _area
    else:
        delta_x = (end - start) / (_n * 1.0)
        x1 = start + n * delta_x
        x2 = start + (n - 1) * delta_x
        rect_height = (f(x1) + f(x2)) / 2.0
        rect_area = rect_height * delta_x
        _area += rect_area
        return trapezoidal(f, start, end, n - 1, _area, _n)

def midpoint(f, start, end, n, _area=0, _n=0):
    if _n is 0:
        return midpoint(f, start, end, n, _area, n)    
    elif n is 0:
        return _area
    else:
        delta_x = (end - start) / (_n * 1.0)
        xi = start + n * delta_x - delta_x / 2
        rect_height = (f(xi))
        rect_area = rect_height * delta_x
        _area += rect_area
        return midpoint(f, start, end, n - 1, _area, _n)

def simpson(f, start, end, n, _area=0, _n=0):
    if n % 2 is 1:
        print "Error: n must be even"
    elif _n is 0:
        return simpson(f, start, end, n, _area, n)    
    elif n is 0:
        return _area
    else:
        delta_x = (end - start) / (_n * 1.0)
        x0 = start + (n) * delta_x
        x1 = start + (n - 1) * delta_x
        x2 = start + (n - 2) * delta_x
        rect_area = (delta_x / 3) * ((f(x0)) + 4 * (f(x1)) + (f(x2)))
        _area += rect_area
        return simpson(f, start, end, n - 2, _area, _n)

def all_approx(f, start, end, n, _area=0, _n=0):
    print "trapezoidal: " + str(trapezoidal(f, start, end, n))
    print "midpoint:    " + str(midpoint(f, start, end, n))
    print "simpson:     " + str(simpson(f, start, end, n))