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import math

from typing import Callable, Union

def line_length(

fnc: Callable[[Union[int, float]], Union[int, float]],

x_start: Union[int, float],

x_end: Union[int, float],

steps: int = 100,

) -> float:

"""

Approximates the arc length of a line segment by treating the curve as a

sequence of linear lines and summing their lengths

:param fnc: a function which defines a curve

:param x_start: left end point to indicate the start of line segment

:param x_end: right end point to indicate end of line segment

:param steps: an accuracy gauge; more steps increases accuracy

:return: a float representing the length of the curve

>>> def f(x):

... return x

>>> f"{line_length(f, 0, 1, 10):.6f}"

'1.414214'

>>> def f(x):

... return 1

>>> f"{line_length(f, -5.5, 4.5):.6f}"

'10.000000'

>>> def f(x):

... return math.sin(5 * x) + math.cos(10 * x) + x * x/10

>>> f"{line_length(f, 0.0, 10.0, 10000):.6f}"

'69.534930'

"""

x1 = x_start

fx1 = fnc(x_start)

length = 0.0

for i in range(steps):

# Approximates curve as a sequence of linear lines and sums their length

x2 = (x_end - x_start) / steps + x1

fx2 = fnc(x2)

length += math.hypot(x2 - x1, fx2 - fx1)

# Increment step

x1 = x2

fx1 = fx2

return length

if __name__ == "__main__":

def f(x):

return math.sin(10 * x)

print("f(x) = sin(10 * x)")

print("The length of the curve from x = -10 to x = 10 is:")

i = 10

while i <= 100000:

print(f"With {i} steps: {line_length(f, -10, 10, i)}")

i *= 10

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