





from __future__ import annotations

from collections.abc import Callable




def     __UpperCamelCase						(_SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE = 100	,		)       ->     float:
  lowercase__			    =						x_start
  lowercase__			    =						fnc(_SCREAMING_SNAKE_CASE	)
  lowercase__			    =						0.0
  for _ in range(_SCREAMING_SNAKE_CASE	):
    # Approximates small segments of curve as linear and solve
    # for trapezoidal area
    lowercase__			    =						(x_end - x_start) / steps + xa
    lowercase__			    =						fnc(_SCREAMING_SNAKE_CASE	)
    area += abs(fxa + fxa	) * (xa - xa) / 2
    # Increment step
    lowercase__			    =						xa
    lowercase__			    =						fxa
  return area


if __name__ == "__main__":




      def     __UpperCamelCase						(_SCREAMING_SNAKE_CASE	)       ->     Union[str, Any]:
        return x**3 + x**2

      print("""f(x) = x^3 + x^2""")
      print("""The area between the curve, x = -5, x = 5 and the x axis is:""")
      lowercase_					     =	10
      while i <= 100_000:
            print(f'''with {i} steps: {trapezoidal_area(f, -5, 5, i)}''')
            i *= 10









from __future__ import annotations

import math
from collections.abc import Callable




def     __UpperCamelCase						(_SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE	,		_SCREAMING_SNAKE_CASE = 100	,		)       ->     float:
  lowercase__			    =						x_start
  lowercase__			    =						fnc(_SCREAMING_SNAKE_CASE	)
  lowercase__			    =						0.0

  for _ in range(_SCREAMING_SNAKE_CASE	):
    # Approximates curve as a sequence of linear lines and sums their length
    lowercase__			    =						(x_end - x_start) / steps + xa
    lowercase__			    =						fnc(_SCREAMING_SNAKE_CASE	)
    length += math.hypot(xa - xa	,		fxa - fxa	)

    # Increment step
    lowercase__			    =						xa
    lowercase__			    =						fxa

  return length


if __name__ == "__main__":




      def     __UpperCamelCase						(_SCREAMING_SNAKE_CASE	)       ->     str:
        return math.sin(10 * x	)

      print("""f(x) = sin(10 * x)""")
      print("""The length of the curve from x = -10 to x = 10 is:""")
      lowercase_					     =	10
      while i <= 100_000:
            print(f'''With {i} steps: {line_length(f, -10, 10, i)}''')
            i *= 10


