import random

# MC integration 1
def compute_pi1(N):
    step = 0
    sum = 0

    while step < N:
        x = random.random()
        y = random.random()

        if x*x + y*y < 1.0:
            sum +=  1
        step += 1

    return (4.0 * sum) / N

# MC integration 2
def compute_pi2(N):
    step = 0
    sum = 0.0

    while step < N:
        x = random.random()
        sum += (1.0 - x*x) ** 0.5
        step += 1

    return 4.0 * sum / N

# Numerical integration 3
def compute_pi3(N):
    x = 0.0
    sum = 0.0

    while x < 1.0:
        sum += (1.0 - x*x) ** 0.5
        x += 1.0/N

    return 4.0 * sum / N