Delete arithmetic_analysis/ directory and relocate its contents (#10824)

* Remove eval from arithmetic_analysis/newton_raphson.py

* Relocate contents of arithmetic_analysis/

Delete the arithmetic_analysis/ directory and relocate its files because
the purpose of the directory was always ill-defined. "Arithmetic
analysis" isn't a field of math, and the directory's files contained
algorithms for linear algebra, numerical analysis, and physics.

Relocated the directory's linear algebra algorithms to linear_algebra/,
its numerical analysis algorithms to a new subdirectory called
maths/numerical_analysis/, and its single physics algorithm to physics/.

* updating DIRECTORY.md

---------

Co-authored-by: github-actions <${GITHUB_ACTOR}@users.noreply.github.com>

maths/numerical_analysis/numerical_integration.py

@@ -0,0 +1,65 @@
+"""
+Approximates the area under the curve using the trapezoidal rule
+"""
+from __future__ import annotations
+
+from collections.abc import Callable
+
+
+def trapezoidal_area(
+    fnc: Callable[[float], float],
+    x_start: float,
+    x_end: 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 _ 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