#### The Rectangular Method

C
{
"cells": [
{
"cell_type": "markdown",
"source": [
"The rectangle method (also called the midpoint rule) is the simplest method in Mathematics used to compute an approximation of a definite integral."
]
},
{
"cell_type": "markdown",
"source": [
"Let's check this method for the next function: $$f(x) = ({e^x / 2})*(cos(x)-sin(x))$$ with $\\varepsilon = 0.001$"
]
},
{
"cell_type": "code",
"execution_count": 1,
"outputs": [],
"source": [
"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"
]
},
{
"cell_type": "markdown",
"source": [
"## Some input data"
]
},
{
"cell_type": "code",
"execution_count": 2,
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"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"
]
}
],
"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)"
]
},
{
"cell_type": "code",
"execution_count": null,
"outputs": [],
"source": []
}
],
"kernelspec": {
"display_name": "Python 3",
"language": "python",
"name": "python3"
},
"language_info": {
"codemirror_mode": {
"name": "ipython",
"version": 3
},
"file_extension": ".py",
"mimetype": "text/x-python",
"name": "python",
"nbconvert_exporter": "python",
"pygments_lexer": "ipython3",
"version": "3.7.6"
}
},
"nbformat": 4,
"nbformat_minor": 4
}


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 &lt; 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&lt;b:

s1=s1+f(i)*z
i=i+z
i=a
while i&lt;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  