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