The Algorithms
The Rectangular Method
C
The rectangle method (also called the midpoint rule) is the simplest method in Mathematics used to compute an approximation of a definite integral.
Let's check this method for the next function: $$f(x) = ({e^x / 2})*(cos(x)-sin(x))$$ with $\varepsilon = 0.001$
import math
import numpy as np
def integration(a,b,n):
h = (b-a)/n
r = f(a) + f(b)
i = 1
while i < n:
x = a + i*h
r = r + 4 * f(x)
i = i + 1
x = a + i * h
r = r +2*f(x)
i = i + 1
r = r * h / 3
print("Result: ", r)
def rectangles(a,b,n):
z = (b-a)/n
i = a
s1=0
s2=0
while i<b:
s1=s1+f(i)*z
i=i+z
i=a
while i<b:
i=i+z
s2=s2+f(i)*z
print('Result of formula of the left rectangles: ',s1)
print('Result of formula of the left rectangles: ',s2)
Some input data
def f(x):
return (math.e**x / 2)*(math.cos(x)-math.sin(x))
n = 4
a = 2.
b = 3.
Si = []
integration(a,b,n)
rectangles(a,b,n)Result: -10.297317477276613
Result of formula of the left rectangles: -7.576924395545134
Result of formula of the left rectangles: -9.192576890365931
