code: 
  
  
  
  
  
  
  '''simple docstring'''
  
  
  
  
  
  from __future__ import annotations
  
  from collections.abc import Callable
  
  
  
  
  
  def    _lowerCamelCase	(	__lowerCamelCase   ,       __lowerCamelCase   ,       __lowerCamelCase   ,       __lowerCamelCase = 100   ,       )   -> float:
  
       '''simple docstring'''
       UpperCAmelCase__			:     Union[str, Any]										= x_start
       UpperCAmelCase__			:     Any										= fnc(_UpperCAmelCase					)
       UpperCAmelCase__			:     Optional[Any]										= 0.0
  
       for _ in range(_UpperCAmelCase					):
            # Approximates small segments of curve as linear and solve
            # for trapezoidal area
            UpperCAmelCase__			:     Tuple										= (x_end - x_start) / steps + xa
            UpperCAmelCase__			:     Dict										= fnc(_UpperCAmelCase					)
            area += abs(fxa + fxa					) * (xa - xa) / 2
  
            # Increment step
            UpperCAmelCase__			:     List[str]										= xa
            UpperCAmelCase__			:     List[str]										= fxa
       return area
  
  
  if __name__ == "__main__":
  
  
  
  
  
        def    _lowerCamelCase	(	__lowerCamelCase					)   -> Optional[int]:
  
             '''simple docstring'''
             return x**3
  
        print("""f(x) = x^3""")
        print("""The area between the curve, x = -10, x = 10 and the x axis is:""")
        SCREAMING_SNAKE_CASE__		:       Tuple   							=							10
        while i <= 10_00_00:
              SCREAMING_SNAKE_CASE__		:       Optional[int]   							=							trapezoidal_area(f, -5, 5, i)
              print(f'''with {i} steps: {area}''')
              i *= 10
  
  
  
  
  
  
  
style_context: 
  
  
  
  
  
  
  import random
  
  
  
  
  
  
  
  class 							UpperCAmelCase_     :
  
  
  
  
       def __init__(							self  ,	_lowerCAmelCase  ,	_lowerCAmelCase		):
            UpperCAmelCase__			:     Union[str, Any]										= x
            UpperCAmelCase__			:     Union[str, Any]										= y
  
  
  
  
       def      __UpperCAmelCase     (							self		):
            return (self.x**2 + self.y**2) <= 1
  
  
  
  
       @classmethod
       def      __UpperCAmelCase     (							cls		):
            return cls(x=random.random()  ,	y=random.random()		)
  
  
  
  
  
  def    _lowerCamelCase	(	__lowerCamelCase					)   -> float:
  
       '''simple docstring'''
       if number_of_simulations < 1:
            raise ValueError("""At least one simulation is necessary to estimate PI."""					)
  
       UpperCAmelCase__			:     Dict										= 0
       for _ in range(__lowerCamelCase					):
            UpperCAmelCase__			:     Union[str, Any]										= Point.random_unit_square()
  
            if random_point.is_in_unit_circle():
                 number_in_unit_circle += 1
  
       return 4 * number_in_unit_circle / number_of_simulations
  
  
  if __name__ == "__main__":
        # import doctest
  
        # doctest.testmod()
        from math import pi
  
        SCREAMING_SNAKE_CASE__		:       List[str]   							=							"""Please enter the desired number of Monte Carlo simulations: """
        SCREAMING_SNAKE_CASE__		:       int   							=							estimate_pi(int(input(prompt).strip()))
        print(f'''An estimate of PI is {my_pi} with an error of {abs(my_pi - pi)}''')
  
  
  
  
  
  
  