#### Numerical Integration

```"""
Approximates the area under the curve using the trapezoidal rule
"""
from __future__ import annotations

from typing import Callable

def trapezoidal_area(
fnc: Callable[[int | float], int | float],
x_start: int | float,
x_end: int | float,
steps: int = 100,
) -> float:

"""
Treats curve as a collection of linear lines and sums the area of the
trapezium shape they form
: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 the accuracy
:return: a float representing the length of the curve

>>> def f(x):
...    return 5
>>> '%.3f' % trapezoidal_area(f, 12.0, 14.0, 1000)
'10.000'

>>> def f(x):
...    return 9*x**2
>>> '%.4f' % trapezoidal_area(f, -4.0, 0, 10000)
'192.0000'

>>> '%.4f' % trapezoidal_area(f, -4.0, 4.0, 10000)
'384.0000'
"""
x1 = x_start
fx1 = fnc(x_start)
area = 0.0

for i in range(steps):

# Approximates small segments of curve as linear and solve
# for trapezoidal area
x2 = (x_end - x_start) / steps + x1
fx2 = fnc(x2)
area += abs(fx2 + fx1) * (x2 - x1) / 2

# Increment step
x1 = x2
fx1 = fx2
return area

if __name__ == "__main__":

def f(x):
return x**3

print("f(x) = x^3")
print("The area between the curve, x = -10, x = 10 and the x axis is:")
i = 10
while i <= 100000:
area = trapezoidal_area(f, -5, 5, i)
print(f"with {i} steps: {area}")
i *= 10
```  