





'''simple docstring'''


from itertools import count
from math import asin, pi, sqrt




def       A							(      _UpperCAmelCase   :				float					)  ->				float:


  '''simple docstring'''




  return (
      (1 - 2 * point) * sqrt(point - point**2					) + 2 * point + asin(sqrt(1 - point					)					)
  ) / 4




def       A							(      _UpperCAmelCase   :				int					)  ->				float:


  '''simple docstring'''




  __lowerCAmelCase		:    Optional[int]			     =  (circles_number + 1 - sqrt(2 * circles_number					)) / (
      2 * (circles_number**2 + 1)
  )
  __lowerCAmelCase		:    List[Any]			     =  circles_number * intersection_y

  __lowerCAmelCase		:    Optional[Any]			     =  intersection_x * intersection_y / 2
  __lowerCAmelCase		:    int			     =  circle_bottom_arc_integral(
      1 / 2					) - circle_bottom_arc_integral(_UpperCAmelCase					)

  return triangle_area + concave_region_area




def       A							(      _UpperCAmelCase   :				float = 1 / 1_0_0_0					)  ->				int:


  '''simple docstring'''




  __lowerCAmelCase		:    int			     =  (1 - pi / 4) / 4

  for n in count(1					):
    if concave_triangle_area(_UpperCAmelCase					) / l_section_area < fraction:
      return n

  return -1


if __name__ == "__main__":
    print(F'''{solution() = }''')












'''simple docstring'''


from __future__ import annotations

from collections.abc import Callable




def       A							(      _UpperCAmelCase   :				Callable[[int | float], int | float]						,_UpperCAmelCase   :				int | float						,_UpperCAmelCase   :				int | float						,_UpperCAmelCase   :				int = 1_0_0						,)  ->				float:


  '''simple docstring'''




  __lowerCAmelCase		:    Dict			     =  x_start
  __lowerCAmelCase		:    Tuple			     =  fnc(_UpperCAmelCase					)
  __lowerCAmelCase		:    Optional[int]			     =  0.0

  for _ in range(_UpperCAmelCase					):
    # Approximates small segments of curve as linear and solve
    # for trapezoidal area
    __lowerCAmelCase		:    Optional[int]			     =  (x_end - x_start) / steps + xa
    __lowerCAmelCase		:    Optional[Any]			     =  fnc(_UpperCAmelCase					)
    area += abs(fxa + fxa					) * (xa - xa) / 2

    # Increment step
    __lowerCAmelCase		:    Optional[Any]			     =  xa
    __lowerCAmelCase		:    List[Any]			     =  fxa
  return area


if __name__ == "__main__":




    def       A							(      _UpperCAmelCase   :				List[Any]					)  ->				List[Any]:


      '''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:")
    A_				     =							10
    while i <= 10_00_00:
        A_				     =							trapezoidal_area(f, -5, 5, i)
        print(F'''with {i} steps: {area}''')
        i *= 10





