From 3042702d04eebfc9a1a545a076b2e6c764e17480 Mon Sep 17 00:00:00 2001
From: Sharan Krishnan <58838321+shrabian@users.noreply.github.com>
Date: Sun, 19 Jan 2020 15:21:12 +1100
Subject: [PATCH] Area Under a Curve Algorithm (#1701)

* A recursive insertion sort

* added doctests and typehints

* Added arc length and numerical integration calculators

* fixed doc test

* Fixed some conversion errors

* Fixed some commenting

* Deleted numerical integration to allow 1 file per push

* Changed string formatting method

* Added program to calculate trapezoidal area under curve

* Deleted files ensure 1 pull request per file

* file name changed

* Update area_under_curve.py

Co-authored-by: Christian Clauss <cclauss@me.com>
---
 maths/area_under_curve.py | 55 +++++++++++++++++++++++++++++++++++++++
 1 file changed, 55 insertions(+)
 create mode 100644 maths/area_under_curve.py

diff --git a/maths/area_under_curve.py b/maths/area_under_curve.py
new file mode 100644
index 000000000..d05e9e4ae
--- /dev/null
+++ b/maths/area_under_curve.py
@@ -0,0 +1,55 @@
+"""
+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