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62 lines
1.6 KiB
Python
62 lines
1.6 KiB
Python
"""
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Approximates the area under the curve using the trapezoidal rule
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"""
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from __future__ import annotations
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from collections.abc import Callable
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def trapezoidal_area(
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fnc: Callable[[float], float],
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x_start: float,
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x_end: float,
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steps: int = 100,
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) -> float:
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"""
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Treats curve as a collection of linear lines and sums the area of the
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trapezium shape they form
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:param fnc: a function which defines a curve
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:param x_start: left end point to indicate the start of line segment
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:param x_end: right end point to indicate end of line segment
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:param steps: an accuracy gauge; more steps increases the accuracy
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:return: a float representing the length of the curve
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>>> def f(x):
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... return 5
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>>> f"{trapezoidal_area(f, 12.0, 14.0, 1000):.3f}"
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'10.000'
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>>> def f(x):
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... return 9*x**2
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>>> f"{trapezoidal_area(f, -4.0, 0, 10000):.4f}"
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'192.0000'
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>>> f"{trapezoidal_area(f, -4.0, 4.0, 10000):.4f}"
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'384.0000'
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"""
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x1 = x_start
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fx1 = fnc(x_start)
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area = 0.0
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for _ in range(steps):
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# Approximates small segments of curve as linear and solve
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# for trapezoidal area
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x2 = (x_end - x_start) / steps + x1
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fx2 = fnc(x2)
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area += abs(fx2 + fx1) * (x2 - x1) / 2
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# Increment step
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x1 = x2
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fx1 = fx2
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return area
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if __name__ == "__main__":
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def f(x):
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return x**3 + x**2
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print("f(x) = x^3 + x^2")
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print("The area between the curve, x = -5, x = 5 and the x axis is:")
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i = 10
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while i <= 100000:
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print(f"with {i} steps: {trapezoidal_area(f, -5, 5, i)}")
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i *= 10
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