#### Line Length

```from __future__ import annotations

import math
from typing import Callable

def line_length(
fnc: Callable[[int | float], int | float],
x_start: int | float,
x_end: 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
```  