# Calculating area under the curve using Monte Carlo method
# FB - 201006137

import math
import random

# define any function here!
def f(x):
    return math.sqrt(1.0 - x * x)

# define any xmin-xmax interval here! (xmin < xmax)
xmin = -1.0
xmax = 1.0

# find ymin-ymax
numSteps = 1000000 # bigger the better but slower!
ymin = f(xmin)
ymax = ymin
for i in range(numSteps):
    x = xmin + (xmax - xmin) * float(i) / numSteps
    y = f(x)
    if y < ymin: ymin = y
    if y > ymax: ymax = y

# Monte Carlo
rectArea = (xmax - xmin) * (ymax - ymin)
numPoints = 1000000 # bigger the better but slower!
ctr = 0
for j in range(numPoints):
    x = xmin + (xmax - xmin) * random.random()
    y = ymin + (ymax - ymin) * random.random()
    if f(x) > 0 and y > 0 and y <= f(x):
        ctr += 1
    if f(x) < 0 and y < 0 and y >= f(x):
        ctr += 1

fnArea = rectArea * float(ctr) / numPoints
print "Area under the curve = " + str(fnArea)