numerical integration Algorithm
In analysis, numerical integration comprises a wide family of algorithms for calculate the numerical value of a definite integral, and by extension, the term is also sometimes used to describe the numerical solution of differential equations. Some writers refer to numerical integration over more than one dimension as cubature; others take quadrature to include higher-dimensional integration. In medieval Europe the quadrature meant calculation of area by any method. quadrature is a historical mathematical term that means calculate area. Mathematicians of Ancient Greece, according to the Pythagorean doctrine, understand calculation of area as the procedure of constructing geometrically a square have the same area (squaring).
"""
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
>>> '%.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("with {} steps: {}".format(i, area))
i *= 10
