numerical_integration.py

"""
Approximates the area under the curve using the trapezoidal rule
"""

from __future__ importannotations

fromcollections.abcimportCallable


deftrapezoidal_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_inrange(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
returnarea


if__name__=="__main__":

deff(x):
returnx**3

print("f(x) = x^3")
print("The area between the curve, x = -10, x = 10 and the x axis is:")
i=10
whilei<=100000:
area=trapezoidal_area(f, -5, 5, i)
print(f"with {i} steps: {area}")
i*=10