

'''simple docstring'''

def  A_	(			snake_case   ,							snake_case			):
  _enforce_args(__UpperCAmelCase   ,							__UpperCAmelCase			)
  if n == 0:
    return 0
  SCREAMING_SNAKE_CASE:Tuple       =       float("-inf"			)
  for i in range(1   ,							n + 1			):
    SCREAMING_SNAKE_CASE:List[str]       =       max(
        __UpperCAmelCase   ,							prices[i - 1] + naive_cut_rod_recursive(n - i   ,							__UpperCAmelCase			)			)

  return max_revue





def  A_	(			snake_case   ,							snake_case			):
  _enforce_args(__UpperCAmelCase   ,							__UpperCAmelCase			)
  SCREAMING_SNAKE_CASE:int       =       [float("-inf"			) for _ in range(n + 1			)]
  return _top_down_cut_rod_recursive(__UpperCAmelCase   ,							__UpperCAmelCase   ,							__UpperCAmelCase			)





def  A_	(			snake_case   ,							snake_case   ,							snake_case			):







  if max_rev[n] >= 0:
    return max_rev[n]
  elif n == 0:
    return 0
  else:
    SCREAMING_SNAKE_CASE:Optional[int]       =       float("-inf"			)
    for i in range(1   ,							n + 1			):
      SCREAMING_SNAKE_CASE:Dict       =       max(
          __UpperCAmelCase   ,							prices[i - 1] + _top_down_cut_rod_recursive(n - i   ,							__UpperCAmelCase   ,							__UpperCAmelCase			)   ,							)

    SCREAMING_SNAKE_CASE:str       =       max_revenue

  return max_rev[n]





def  A_	(			snake_case   ,							snake_case			):
  _enforce_args(__UpperCAmelCase   ,							__UpperCAmelCase			)

  # length(max_rev) = n + 1, to accommodate for the revenue obtainable from a rod of
  # length 0.
  SCREAMING_SNAKE_CASE:List[Any]       =       [float("-inf"			) for _ in range(n + 1			)]
  SCREAMING_SNAKE_CASE:Dict       =       0

  for i in range(1   ,							n + 1			):
    SCREAMING_SNAKE_CASE:str       =       max_rev[i]
    for j in range(1   ,							i + 1			):
      SCREAMING_SNAKE_CASE:Dict       =       max(__UpperCAmelCase   ,							prices[j - 1] + max_rev[i - j]			)

    SCREAMING_SNAKE_CASE:Union[str, Any]       =       max_revenue_i

  return max_rev[n]





def  A_	(			snake_case   ,							snake_case			):







  if n < 0:
    SCREAMING_SNAKE_CASE:Any       =       F'''n must be greater than or equal to 0. Got n = {n}'''
    raise ValueError(__UpperCAmelCase			)

  if n > len(__UpperCAmelCase			):
    SCREAMING_SNAKE_CASE:Optional[Any]       =       (
        "Each integral piece of rod must have a corresponding price. "
        F'''Got n = {n} but length of prices = {len(__UpperCAmelCase			)}'''
    )
    raise ValueError(__UpperCAmelCase			)





def  A_	(			):
  SCREAMING_SNAKE_CASE:int       =       [6, 10, 12, 15, 20, 23]
  SCREAMING_SNAKE_CASE:List[str]       =       len(__UpperCAmelCase			)

  # the best revenue comes from cutting the rod into 6 pieces, each
  # of length 1 resulting in a revenue of 6 * 6 = 36.
  SCREAMING_SNAKE_CASE:Union[str, Any]       =       36

  SCREAMING_SNAKE_CASE:Tuple       =       top_down_cut_rod(__UpperCAmelCase   ,							__UpperCAmelCase			)
  SCREAMING_SNAKE_CASE:int       =       bottom_up_cut_rod(__UpperCAmelCase   ,							__UpperCAmelCase			)
  SCREAMING_SNAKE_CASE:Optional[int]       =       naive_cut_rod_recursive(__UpperCAmelCase   ,							__UpperCAmelCase			)

  assert expected_max_revenue == max_rev_top_down
  assert max_rev_top_down == max_rev_bottom_up
  assert max_rev_bottom_up == max_rev_naive


if __name__ == "__main__":
    main()













'''simple docstring'''

def     __magic_name__							(				__UpperCAmelCase				,						__UpperCAmelCase   )    ->       str:

   '''simple docstring'''







   __SCREAMING_SNAKE_CASE   				=							(boundary[1] - boundary[0]) / steps
   __SCREAMING_SNAKE_CASE   				=							boundary[0]
   __SCREAMING_SNAKE_CASE   				=							boundary[1]
   __SCREAMING_SNAKE_CASE   				=							make_points(__UpperCAmelCase				,						__UpperCAmelCase				,						__UpperCAmelCase   )
   __SCREAMING_SNAKE_CASE   				=							0.0
   y += (h / 2.0) * f(__UpperCAmelCase   )
   for i in x_i:
      # print(i)
      y += h * f(__UpperCAmelCase   )
   y += (h / 2.0) * f(__UpperCAmelCase   )
   return y

def     __magic_name__							(				__UpperCAmelCase				,						__UpperCAmelCase				,						__UpperCAmelCase   )    ->       Optional[Any]:

   '''simple docstring'''







   __SCREAMING_SNAKE_CASE   				=							a + h
   while x < (b - h):
      yield x
      __SCREAMING_SNAKE_CASE   				=							x + h

def     __magic_name__							(				__UpperCAmelCase   )    ->       int:  # enter your function here

   '''simple docstring'''







   __SCREAMING_SNAKE_CASE   				=							(x - 0) * (x - 0)
   return y

def     __magic_name__							(				)    ->       Tuple:

   '''simple docstring'''







   __SCREAMING_SNAKE_CASE   				=							0.0  # Lower bound of integration
   __SCREAMING_SNAKE_CASE   				=							1.0  # Upper bound of integration
   __SCREAMING_SNAKE_CASE   				=							1_0.0  # define number of steps or resolution
   __SCREAMING_SNAKE_CASE   				=							[a, b]  # define boundary of integration
   __SCREAMING_SNAKE_CASE   				=							method_a(__UpperCAmelCase				,						__UpperCAmelCase   )
   print(f"""y = {y}"""   )


if __name__ == "__main__":
    main()





