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"""

Approximates the area under the curve using the trapezoidal rule

"""

from typing import Callable, Union

def trapezoidal_area(

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

x_start: Union[int, float],

x_end: Union[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

>>> f"{trapezoidal_area(f, 12.0, 14.0, 1000):.3f}"

'10.000'

>>> def f(x):

... return 9*x**2

>>> f"{trapezoidal_area(f, -4.0, 0, 10000):.4f}"

'192.0000'

>>> f"{trapezoidal_area(f, -4.0, 4.0, 10000):.4f}"

'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 + x ** 2

print("f(x) = x^3 + x^2")

print("The area between the curve, x = -5, x = 5 and the x axis is:")

i = 10

while i <= 100000:

print(f"with {i} steps: {trapezoidal_area(f, -5, 5, i)}")

i *= 10

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