SIMPSONS-RULE.ipynb

{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from numpy import *\n",
    "from sympy import *\n",
    "from sympy.abc import r\n",
    "\n",
    "#Lets define the Simpson Rule constants in the formula\n",
    "A =0      #initial coordinate\n",
    "B=3       #Final coordinate\n",
    "N=397       #Number of rectangles (N must be ODD), note that the higher the value of N, the more accurate the final answer\n",
    "h = (B-A)/(N-1) #width of each rectangle"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "#####Lets start writing a function that includes that calls the function to be integrated\n",
    "\n",
    "def f(x):\n",
    "    f = x*x\n",
    "    print('x f(x) =',x,x*x)\n",
    "    return (f)\n",
    "##########################################################No need to change this part of the code-start\n",
    "def Simps_ih(i,h):\n",
    "    if ((i ==1) or (i==N)):\n",
    "        width =h/3\n",
    "        return (width)\n",
    "    elif (i%2==0):\n",
    "        width = (4*h)/3\n",
    "        return (width)\n",
    "    else:\n",
    "        width = (2*h)/3\n",
    "        return (width)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x f(x) = 0.0 0.0\n",
      "x f(x) = 0.007575757575757576 5.739210284664831e-05\n",
      "x f(x) = 0.015151515151515152 0.00022956841138659323\n",
      "x f(x) = 0.022727272727272728 0.0005165289256198347\n",
      "x f(x) = 0.030303030303030304 0.0009182736455463729\n",
      "x f(x) = 0.03787878787878788 0.0014348025711662076\n",
      "x f(x) = 0.045454545454545456 0.002066115702479339\n",
      "x f(x) = 0.05303030303030303 0.002812213039485767\n",
      "x f(x) = 0.06060606060606061 0.0036730945821854917\n",
      "x f(x) = 0.06818181818181818 0.004648760330578512\n",
      "x f(x) = 0.07575757575757576 0.005739210284664831\n",
      "x f(x) = 0.08333333333333334 0.006944444444444446\n",
      "x f(x) = 0.09090909090909091 0.008264462809917356\n",
      "x f(x) = 0.09848484848484848 0.009699265381083563\n",
      "x f(x) = 0.10606060606060606 0.011248852157943069\n",
      "x f(x) = 0.11363636363636365 0.01291322314049587\n",
      "x f(x) = 0.12121212121212122 0.014692378328741967\n",
      "x f(x) = 0.12878787878787878 0.01658631772268136\n",
      "x f(x) = 0.13636363636363635 0.018595041322314047\n",
      "x f(x) = 0.14393939393939395 0.02071854912764004\n",
      "x f(x) = 0.15151515151515152 0.022956841138659322\n",
      "x f(x) = 0.1590909090909091 0.0253099173553719\n",
      "x f(x) = 0.16666666666666669 0.027777777777777783\n",
      "x f(x) = 0.17424242424242425 0.030360422405876955\n",
      "x f(x) = 0.18181818181818182 0.03305785123966942\n",
      "x f(x) = 0.1893939393939394 0.035870064279155185\n",
      "x f(x) = 0.19696969696969696 0.03879706152433425\n",
      "x f(x) = 0.20454545454545456 0.04183884297520662\n",
      "x f(x) = 0.21212121212121213 0.044995408631772274\n",
      "x f(x) = 0.2196969696969697 0.04826675849403122\n",
      "x f(x) = 0.2272727272727273 0.05165289256198348\n",
      "x f(x) = 0.23484848484848486 0.05515381083562902\n",
      "x f(x) = 0.24242424242424243 0.05876951331496787\n",
      "x f(x) = 0.25 0.0625\n",
      "x f(x) = 0.25757575757575757 0.06634527089072544\n",
      "x f(x) = 0.26515151515151514 0.07030532598714416\n",
      "x f(x) = 0.2727272727272727 0.07438016528925619\n",
      "x f(x) = 0.28030303030303033 0.07856978879706154\n",
      "x f(x) = 0.2878787878787879 0.08287419651056016\n",
      "x f(x) = 0.29545454545454547 0.08729338842975208\n",
      "x f(x) = 0.30303030303030304 0.09182736455463729\n",
      "x f(x) = 0.3106060606060606 0.0964761248852158\n",
      "x f(x) = 0.3181818181818182 0.1012396694214876\n",
      "x f(x) = 0.32575757575757575 0.1061179981634527\n",
      "x f(x) = 0.33333333333333337 0.11111111111111113\n",
      "x f(x) = 0.34090909090909094 0.11621900826446283\n",
      "x f(x) = 0.3484848484848485 0.12144168962350782\n",
      "x f(x) = 0.3560606060606061 0.12677915518824612\n",
      "x f(x) = 0.36363636363636365 0.1322314049586777\n",
      "x f(x) = 0.3712121212121212 0.13779843893480256\n",
      "x f(x) = 0.3787878787878788 0.14348025711662074\n",
      "x f(x) = 0.38636363636363635 0.14927685950413222\n",
      "x f(x) = 0.3939393939393939 0.155188246097337\n",
      "x f(x) = 0.40151515151515155 0.1612144168962351\n",
      "x f(x) = 0.4090909090909091 0.16735537190082647\n",
      "x f(x) = 0.4166666666666667 0.17361111111111113\n",
      "x f(x) = 0.42424242424242425 0.1799816345270891\n",
      "x f(x) = 0.4318181818181818 0.18646694214876033\n",
      "x f(x) = 0.4393939393939394 0.19306703397612487\n",
      "x f(x) = 0.44696969696969696 0.19978191000918272\n",
      "x f(x) = 0.4545454545454546 0.20661157024793392\n",
      "x f(x) = 0.46212121212121215 0.21355601469237837\n",
      "x f(x) = 0.4696969696969697 0.2206152433425161\n",
      "x f(x) = 0.4772727272727273 0.22778925619834714\n",
      "x f(x) = 0.48484848484848486 0.23507805325987147\n",
      "x f(x) = 0.49242424242424243 0.24248163452708907\n",
      "x f(x) = 0.5 0.25\n",
      "x f(x) = 0.5075757575757576 0.25763314967860423\n",
      "x f(x) = 0.5151515151515151 0.26538108356290174\n",
      "x f(x) = 0.5227272727272727 0.2732438016528925\n",
      "x f(x) = 0.5303030303030303 0.28122130394857664\n",
      "x f(x) = 0.5378787878787878 0.28931359044995403\n",
      "x f(x) = 0.5454545454545454 0.29752066115702475\n",
      "x f(x) = 0.5530303030303031 0.30584251606978885\n",
      "x f(x) = 0.5606060606060607 0.3142791551882462\n",
      "x f(x) = 0.5681818181818182 0.32283057851239677\n",
      "x f(x) = 0.5757575757575758 0.33149678604224064\n",
      "x f(x) = 0.5833333333333334 0.34027777777777785\n",
      "x f(x) = 0.5909090909090909 0.3491735537190083\n",
      "x f(x) = 0.5984848484848485 0.3581841138659321\n",
      "x f(x) = 0.6060606060606061 0.36730945821854916\n",
      "x f(x) = 0.6136363636363636 0.3765495867768595\n",
      "x f(x) = 0.6212121212121212 0.3859044995408632\n",
      "x f(x) = 0.6287878787878788 0.39537419651056016\n",
      "x f(x) = 0.6363636363636364 0.4049586776859504\n",
      "x f(x) = 0.6439393939393939 0.41465794306703396\n",
      "x f(x) = 0.6515151515151515 0.4244719926538108\n",
      "x f(x) = 0.6590909090909091 0.434400826446281\n",
      "x f(x) = 0.6666666666666667 0.44444444444444453\n",
      "x f(x) = 0.6742424242424243 0.4546028466483013\n",
      "x f(x) = 0.6818181818181819 0.4648760330578513\n",
      "x f(x) = 0.6893939393939394 0.4752640036730947\n",
      "x f(x) = 0.696969696969697 0.4857667584940313\n",
      "x f(x) = 0.7045454545454546 0.4963842975206612\n",
      "x f(x) = 0.7121212121212122 0.5071166207529845\n",
      "x f(x) = 0.7196969696969697 0.517963728191001\n",
      "x f(x) = 0.7272727272727273 0.5289256198347108\n",
      "x f(x) = 0.7348484848484849 0.5400022956841138\n",
      "x f(x) = 0.7424242424242424 0.5511937557392103\n",
      "x f(x) = 0.75 0.5625\n",
      "x f(x) = 0.7575757575757576 0.573921028466483\n",
      "x f(x) = 0.7651515151515151 0.5854568411386593\n",
      "x f(x) = 0.7727272727272727 0.5971074380165289\n",
      "x f(x) = 0.7803030303030303 0.6088728191000918\n",
      "x f(x) = 0.7878787878787878 0.620752984389348\n",
      "x f(x) = 0.7954545454545455 0.6327479338842976\n",
      "x f(x) = 0.8030303030303031 0.6448576675849405\n",
      "x f(x) = 0.8106060606060607 0.6570821854912765\n",
      "x f(x) = 0.8181818181818182 0.6694214876033059\n",
      "x f(x) = 0.8257575757575758 0.6818755739210285\n",
      "x f(x) = 0.8333333333333334 0.6944444444444445\n",
      "x f(x) = 0.8409090909090909 0.7071280991735538\n",
      "x f(x) = 0.8484848484848485 0.7199265381083564\n",
      "x f(x) = 0.8560606060606061 0.7328397612488522\n",
      "x f(x) = 0.8636363636363636 0.7458677685950413\n",
      "x f(x) = 0.8712121212121212 0.7590105601469238\n",
      "x f(x) = 0.8787878787878788 0.7722681359044995\n",
      "x f(x) = 0.8863636363636364 0.7856404958677686\n",
      "x f(x) = 0.8939393939393939 0.7991276400367309\n",
      "x f(x) = 0.9015151515151515 0.8127295684113865\n",
      "x f(x) = 0.9090909090909092 0.8264462809917357\n",
      "x f(x) = 0.9166666666666667 0.8402777777777779\n",
      "x f(x) = 0.9242424242424243 0.8542240587695135\n",
      "x f(x) = 0.9318181818181819 0.8682851239669422\n",
      "x f(x) = 0.9393939393939394 0.8824609733700643\n",
      "x f(x) = 0.946969696969697 0.8967516069788798\n",
      "x f(x) = 0.9545454545454546 0.9111570247933886\n",
      "x f(x) = 0.9621212121212122 0.9256772268135905\n",
      "x f(x) = 0.9696969696969697 0.9403122130394859\n",
      "x f(x) = 0.9772727272727273 0.9550619834710744\n",
      "x f(x) = 0.9848484848484849 0.9699265381083563\n",
      "x f(x) = 0.9924242424242424 0.9849058769513315\n",
      "x f(x) = 1.0 1.0\n",
      "x f(x) = 1.0075757575757576 1.0152089072543617\n",
      "x f(x) = 1.0151515151515151 1.030532598714417\n",
      "x f(x) = 1.0227272727272727 1.0459710743801653\n",
      "x f(x) = 1.0303030303030303 1.061524334251607\n",
      "x f(x) = 1.0378787878787878 1.0771923783287418\n",
      "x f(x) = 1.0454545454545454 1.09297520661157\n",
      "x f(x) = 1.053030303030303 1.1088728191000918\n",
      "x f(x) = 1.0606060606060606 1.1248852157943066\n",
      "x f(x) = 1.0681818181818181 1.1410123966942147\n",
      "x f(x) = 1.0757575757575757 1.1572543617998161\n",
      "x f(x) = 1.0833333333333333 1.173611111111111\n",
      "x f(x) = 1.0909090909090908 1.190082644628099\n",
      "x f(x) = 1.0984848484848486 1.206668962350781\n",
      "x f(x) = 1.1060606060606062 1.2233700642791554\n",
      "x f(x) = 1.1136363636363638 1.2401859504132233\n",
      "x f(x) = 1.1212121212121213 1.2571166207529847\n",
      "x f(x) = 1.128787878787879 1.2741620752984393\n",
      "x f(x) = 1.1363636363636365 1.291322314049587\n",
      "x f(x) = 1.143939393939394 1.308597337006428\n",
      "x f(x) = 1.1515151515151516 1.3259871441689626\n",
      "x f(x) = 1.1590909090909092 1.3434917355371903\n",
      "x f(x) = 1.1666666666666667 1.3611111111111114\n",
      "x f(x) = 1.1742424242424243 1.3788452708907255\n",
      "x f(x) = 1.1818181818181819 1.3966942148760333\n",
      "x f(x) = 1.1893939393939394 1.414657943067034\n",
      "x f(x) = 1.196969696969697 1.4327364554637283\n",
      "x f(x) = 1.2045454545454546 1.4509297520661157\n",
      "x f(x) = 1.2121212121212122 1.4692378328741966\n",
      "x f(x) = 1.2196969696969697 1.4876606978879707\n",
      "x f(x) = 1.2272727272727273 1.506198347107438\n",
      "x f(x) = 1.2348484848484849 1.5248507805325988\n",
      "x f(x) = 1.2424242424242424 1.5436179981634528\n",
      "x f(x) = 1.25 1.5625\n",
      "x f(x) = 1.2575757575757576 1.5814967860422406\n",
      "x f(x) = 1.2651515151515151 1.6006083562901745\n",
      "x f(x) = 1.2727272727272727 1.6198347107438016\n",
      "x f(x) = 1.2803030303030303 1.639175849403122\n",
      "x f(x) = 1.2878787878787878 1.6586317722681359\n",
      "x f(x) = 1.2954545454545454 1.6782024793388428\n",
      "x f(x) = 1.303030303030303 1.6978879706152432\n",
      "x f(x) = 1.3106060606060606 1.7176882460973368\n",
      "x f(x) = 1.3181818181818181 1.737603305785124\n",
      "x f(x) = 1.3257575757575757 1.757633149678604\n",
      "x f(x) = 1.3333333333333335 1.7777777777777781\n",
      "x f(x) = 1.340909090909091 1.798037190082645\n",
      "x f(x) = 1.3484848484848486 1.8184113865932052\n",
      "x f(x) = 1.3560606060606062 1.8389003673094586\n",
      "x f(x) = 1.3636363636363638 1.8595041322314052\n",
      "x f(x) = 1.3712121212121213 1.8802226813590452\n",
      "x f(x) = 1.378787878787879 1.9010560146923787\n",
      "x f(x) = 1.3863636363636365 1.9220041322314052\n",
      "x f(x) = 1.393939393939394 1.9430670339761251\n",
      "x f(x) = 1.4015151515151516 1.9642447199265383\n",
      "x f(x) = 1.4090909090909092 1.9855371900826448\n",
      "x f(x) = 1.4166666666666667 2.0069444444444446\n",
      "x f(x) = 1.4242424242424243 2.028466483011938\n",
      "x f(x) = 1.4318181818181819 2.0501033057851243\n",
      "x f(x) = 1.4393939393939394 2.071854912764004\n",
      "x f(x) = 1.446969696969697 2.093721303948577\n",
      "x f(x) = 1.4545454545454546 2.115702479338843\n",
      "x f(x) = 1.4621212121212122 2.137798438934803\n",
      "x f(x) = 1.4696969696969697 2.1600091827364554\n",
      "x f(x) = 1.4772727272727273 2.1823347107438016\n",
      "x f(x) = 1.4848484848484849 2.204775022956841\n",
      "x f(x) = 1.4924242424242424 2.227330119375574\n",
      "x f(x) = 1.5 2.25\n",
      "x f(x) = 1.5075757575757576 2.2727846648301195\n",
      "x f(x) = 1.5151515151515151 2.295684113865932\n",
      "x f(x) = 1.5227272727272727 2.318698347107438\n",
      "x f(x) = 1.5303030303030303 2.341827364554637\n",
      "x f(x) = 1.5378787878787878 2.36507116620753\n",
      "x f(x) = 1.5454545454545454 2.3884297520661155\n",
      "x f(x) = 1.553030303030303 2.411903122130395\n",
      "x f(x) = 1.5606060606060606 2.4354912764003673\n",
      "x f(x) = 1.5681818181818181 2.459194214876033\n",
      "x f(x) = 1.5757575757575757 2.483011937557392\n",
      "x f(x) = 1.5833333333333335 2.506944444444445\n",
      "x f(x) = 1.590909090909091 2.5309917355371905\n",
      "x f(x) = 1.5984848484848486 2.5551538108356295\n",
      "x f(x) = 1.6060606060606062 2.579430670339762\n",
      "x f(x) = 1.6136363636363638 2.6038223140495873\n",
      "x f(x) = 1.6212121212121213 2.628328741965106\n",
      "x f(x) = 1.628787878787879 2.652949954086318\n",
      "x f(x) = 1.6363636363636365 2.6776859504132235\n",
      "x f(x) = 1.643939393939394 2.7025367309458224\n",
      "x f(x) = 1.6515151515151516 2.727502295684114\n",
      "x f(x) = 1.6590909090909092 2.7525826446280997\n",
      "x f(x) = 1.6666666666666667 2.777777777777778\n",
      "x f(x) = 1.6742424242424243 2.80308769513315\n",
      "x f(x) = 1.6818181818181819 2.828512396694215\n",
      "x f(x) = 1.6893939393939394 2.8540518824609737\n",
      "x f(x) = 1.696969696969697 2.8797061524334255\n",
      "x f(x) = 1.7045454545454546 2.9054752066115705\n",
      "x f(x) = 1.7121212121212122 2.931359044995409\n",
      "x f(x) = 1.7196969696969697 2.9573576675849402\n",
      "x f(x) = 1.7272727272727273 2.9834710743801653\n",
      "x f(x) = 1.7348484848484849 3.0096992653810837\n",
      "x f(x) = 1.7424242424242424 3.0360422405876952\n",
      "x f(x) = 1.75 3.0625\n",
      "x f(x) = 1.7575757575757576 3.089072543617998\n",
      "x f(x) = 1.7651515151515151 3.1157598714416896\n",
      "x f(x) = 1.7727272727272727 3.1425619834710745\n",
      "x f(x) = 1.7803030303030303 3.169478879706152\n",
      "x f(x) = 1.7878787878787878 3.1965105601469235\n",
      "x f(x) = 1.7954545454545454 3.2236570247933884\n",
      "x f(x) = 1.803030303030303 3.250918273645546\n",
      "x f(x) = 1.8106060606060606 3.2782943067033976\n",
      "x f(x) = 1.8181818181818183 3.3057851239669427\n",
      "x f(x) = 1.825757575757576 3.3333907254361805\n",
      "x f(x) = 1.8333333333333335 3.3611111111111116\n",
      "x f(x) = 1.840909090909091 3.388946280991736\n",
      "x f(x) = 1.8484848484848486 3.416896235078054\n",
      "x f(x) = 1.8560606060606062 3.4449609733700646\n",
      "x f(x) = 1.8636363636363638 3.473140495867769\n",
      "x f(x) = 1.8712121212121213 3.5014348025711666\n",
      "x f(x) = 1.878787878787879 3.5298438934802574\n",
      "x f(x) = 1.8863636363636365 3.558367768595042\n",
      "x f(x) = 1.893939393939394 3.587006427915519\n",
      "x f(x) = 1.9015151515151516 3.61575987144169\n",
      "x f(x) = 1.9090909090909092 3.6446280991735542\n",
      "x f(x) = 1.9166666666666667 3.6736111111111116\n",
      "x f(x) = 1.9242424242424243 3.702708907254362\n",
      "x f(x) = 1.9318181818181819 3.731921487603306\n",
      "x f(x) = 1.9393939393939394 3.7612488521579435\n",
      "x f(x) = 1.946969696969697 3.7906910009182737\n",
      "x f(x) = 1.9545454545454546 3.8202479338842976\n",
      "x f(x) = 1.9621212121212122 3.8499196510560147\n",
      "x f(x) = 1.9696969696969697 3.879706152433425\n",
      "x f(x) = 1.9772727272727273 3.909607438016529\n",
      "x f(x) = 1.9848484848484849 3.939623507805326\n",
      "x f(x) = 1.9924242424242424 3.9697543617998163\n",
      "x f(x) = 2.0 4.0\n",
      "x f(x) = 2.007575757575758 4.030360422405878\n",
      "x f(x) = 2.015151515151515 4.060835629017447\n",
      "x f(x) = 2.022727272727273 4.091425619834712\n",
      "x f(x) = 2.0303030303030303 4.122130394857668\n",
      "x f(x) = 2.037878787878788 4.152949954086319\n",
      "x f(x) = 2.0454545454545454 4.183884297520661\n",
      "x f(x) = 2.053030303030303 4.214933425160699\n",
      "x f(x) = 2.0606060606060606 4.246097337006428\n",
      "x f(x) = 2.0681818181818183 4.277376033057852\n",
      "x f(x) = 2.0757575757575757 4.308769513314967\n",
      "x f(x) = 2.0833333333333335 4.340277777777779\n",
      "x f(x) = 2.090909090909091 4.37190082644628\n",
      "x f(x) = 2.0984848484848486 4.403638659320478\n",
      "x f(x) = 2.106060606060606 4.435491276400367\n",
      "x f(x) = 2.1136363636363638 4.467458677685951\n",
      "x f(x) = 2.121212121212121 4.499540863177226\n",
      "x f(x) = 2.128787878787879 4.531737832874197\n",
      "x f(x) = 2.1363636363636362 4.564049586776859\n",
      "x f(x) = 2.143939393939394 4.596476124885216\n",
      "x f(x) = 2.1515151515151514 4.6290174471992644\n",
      "x f(x) = 2.159090909090909 4.661673553719009\n",
      "x f(x) = 2.1666666666666665 4.694444444444444\n",
      "x f(x) = 2.1742424242424243 4.727330119375575\n",
      "x f(x) = 2.1818181818181817 4.760330578512396\n",
      "x f(x) = 2.1893939393939394 4.793445821854913\n",
      "x f(x) = 2.1969696969696972 4.826675849403124\n",
      "x f(x) = 2.2045454545454546 4.860020661157025\n",
      "x f(x) = 2.2121212121212124 4.893480257116622\n",
      "x f(x) = 2.2196969696969697 4.92705463728191\n",
      "x f(x) = 2.2272727272727275 4.960743801652893\n",
      "x f(x) = 2.234848484848485 4.994547750229568\n",
      "x f(x) = 2.2424242424242427 5.028466483011939\n",
      "x f(x) = 2.25 5.0625\n",
      "x f(x) = 2.257575757575758 5.096648301193757\n",
      "x f(x) = 2.265151515151515 5.130911386593205\n",
      "x f(x) = 2.272727272727273 5.165289256198348\n",
      "x f(x) = 2.2803030303030303 5.199781910009182\n",
      "x f(x) = 2.287878787878788 5.234389348025712\n",
      "x f(x) = 2.2954545454545454 5.269111570247934\n",
      "x f(x) = 2.303030303030303 5.30394857667585\n",
      "x f(x) = 2.3106060606060606 5.338900367309458\n",
      "x f(x) = 2.3181818181818183 5.373966942148761\n",
      "x f(x) = 2.3257575757575757 5.409148301193755\n",
      "x f(x) = 2.3333333333333335 5.4444444444444455\n",
      "x f(x) = 2.340909090909091 5.479855371900826\n",
      "x f(x) = 2.3484848484848486 5.515381083562902\n",
      "x f(x) = 2.356060606060606 5.55102157943067\n",
      "x f(x) = 2.3636363636363638 5.586776859504133\n",
      "x f(x) = 2.371212121212121 5.622646923783287\n",
      "x f(x) = 2.378787878787879 5.658631772268136\n",
      "x f(x) = 2.3863636363636362 5.694731404958677\n",
      "x f(x) = 2.393939393939394 5.730945821854913\n",
      "x f(x) = 2.4015151515151514 5.767275022956841\n",
      "x f(x) = 2.409090909090909 5.803719008264463\n",
      "x f(x) = 2.4166666666666665 5.840277777777777\n",
      "x f(x) = 2.4242424242424243 5.8769513314967865\n",
      "x f(x) = 2.431818181818182 5.9137396694214885\n",
      "x f(x) = 2.4393939393939394 5.950642791551883\n",
      "x f(x) = 2.4469696969696972 5.987660697887972\n",
      "x f(x) = 2.4545454545454546 6.024793388429752\n",
      "x f(x) = 2.4621212121212124 6.062040863177228\n",
      "x f(x) = 2.4696969696969697 6.099403122130395\n",
      "x f(x) = 2.4772727272727275 6.1368801652892575\n",
      "x f(x) = 2.484848484848485 6.174471992653811\n",
      "x f(x) = 2.4924242424242427 6.21217860422406\n",
      "x f(x) = 2.5 6.25\n",
      "x f(x) = 2.507575757575758 6.287936179981636\n",
      "x f(x) = 2.515151515151515 6.325987144168963\n",
      "x f(x) = 2.522727272727273 6.364152892561984\n",
      "x f(x) = 2.5303030303030303 6.402433425160698\n",
      "x f(x) = 2.537878787878788 6.440828741965107\n",
      "x f(x) = 2.5454545454545454 6.479338842975206\n",
      "x f(x) = 2.553030303030303 6.517963728191002\n",
      "x f(x) = 2.5606060606060606 6.556703397612488\n",
      "x f(x) = 2.5681818181818183 6.59555785123967\n",
      "x f(x) = 2.5757575757575757 6.634527089072543\n",
      "x f(x) = 2.5833333333333335 6.673611111111112\n",
      "x f(x) = 2.590909090909091 6.712809917355371\n",
      "x f(x) = 2.5984848484848486 6.752123507805327\n",
      "x f(x) = 2.606060606060606 6.791551882460973\n",
      "x f(x) = 2.6136363636363638 6.831095041322315\n",
      "x f(x) = 2.621212121212121 6.870752984389347\n",
      "x f(x) = 2.628787878787879 6.910525711662076\n",
      "x f(x) = 2.6363636363636362 6.950413223140496\n",
      "x f(x) = 2.643939393939394 6.99041551882461\n",
      "x f(x) = 2.6515151515151514 7.030532598714416\n",
      "x f(x) = 2.659090909090909 7.070764462809918\n",
      "x f(x) = 2.666666666666667 7.1111111111111125\n",
      "x f(x) = 2.6742424242424243 7.151572543617998\n",
      "x f(x) = 2.681818181818182 7.19214876033058\n",
      "x f(x) = 2.6893939393939394 7.232839761248853\n",
      "x f(x) = 2.6969696969696972 7.273645546372821\n",
      "x f(x) = 2.7045454545454546 7.31456611570248\n",
      "x f(x) = 2.7121212121212124 7.355601469237834\n",
      "x f(x) = 2.7196969696969697 7.39675160697888\n",
      "x f(x) = 2.7272727272727275 7.438016528925621\n",
      "x f(x) = 2.734848484848485 7.479396235078053\n",
      "x f(x) = 2.7424242424242427 7.520890725436181\n",
      "x f(x) = 2.75 7.5625\n",
      "x f(x) = 2.757575757575758 7.604224058769515\n",
      "x f(x) = 2.765151515151515 7.6460629017447195\n",
      "x f(x) = 2.772727272727273 7.688016528925621\n",
      "x f(x) = 2.7803030303030303 7.730084940312213\n",
      "x f(x) = 2.787878787878788 7.7722681359045005\n",
      "x f(x) = 2.7954545454545454 7.814566115702479\n",
      "x f(x) = 2.803030303030303 7.856978879706153\n",
      "x f(x) = 2.8106060606060606 7.899506427915519\n",
      "x f(x) = 2.8181818181818183 7.942148760330579\n",
      "x f(x) = 2.8257575757575757 7.9849058769513315\n",
      "x f(x) = 2.8333333333333335 8.027777777777779\n",
      "x f(x) = 2.840909090909091 8.070764462809917\n",
      "x f(x) = 2.8484848484848486 8.113865932047752\n",
      "x f(x) = 2.856060606060606 8.157082185491277\n",
      "x f(x) = 2.8636363636363638 8.200413223140497\n",
      "x f(x) = 2.871212121212121 8.243859044995409\n",
      "x f(x) = 2.878787878787879 8.287419651056016\n",
      "x f(x) = 2.8863636363636362 8.331095041322314\n",
      "x f(x) = 2.893939393939394 8.374885215794308\n",
      "x f(x) = 2.9015151515151514 8.418790174471992\n",
      "x f(x) = 2.909090909090909 8.462809917355372\n",
      "x f(x) = 2.916666666666667 8.506944444444446\n",
      "x f(x) = 2.9242424242424243 8.551193755739211\n",
      "x f(x) = 2.931818181818182 8.59555785123967\n",
      "x f(x) = 2.9393939393939394 8.640036730945821\n",
      "x f(x) = 2.9469696969696972 8.684630394857669\n",
      "x f(x) = 2.9545454545454546 8.729338842975206\n",
      "x f(x) = 2.9621212121212124 8.77416207529844\n",
      "x f(x) = 2.9696969696969697 8.819100091827364\n",
      "x f(x) = 2.9772727272727275 8.864152892561984\n",
      "x f(x) = 2.984848484848485 8.909320477502296\n",
      "x f(x) = 2.9924242424242427 8.954602846648303\n",
      "x f(x) = 3.0 9.0\n",
      "\n",
      "Total Area = 9.000000000000005\n"
     ]
    }
   ],
   "source": [
    "##Initialising the Area\n",
    "SUM_A = 0.0\n",
    "\n",
    "###Loop for calculating the x coordinates(x0,x1,x2,x3,x4)\n",
    "for i in range(1,N+1):\n",
    "    t = A+ (i-1)*h\n",
    "    #print(t)\n",
    "    #print(f(t))\n",
    "    w = Simps_ih(i,h)\n",
    "    #print(w)\n",
    "    SUM_A = SUM_A + w*f(t)  #This line calculates the total sum of the area by Trapezoidal Rule\n",
    "print('\\nTotal Area =',SUM_A)\n",
    "\n",
    "##########################################################No need to change this part of the code-end\n",
    "\n",
    "#####################Finding exact integral and compare"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "\n",
      "The exact area by integration of a function is = 9.00000000000000\n",
      "\n",
      "The error made by integrating a fuction using... \n",
      "Simpson rule of 397 rectangles and exact solution is = 5.32907051820075E-15\n",
      "\n"
     ]
    }
   ],
   "source": [
    "\n",
    "F = Integral(r**2,(r,0,3))\n",
    "F.as_sum(4,method ='midpoint')\n",
    "A_ex = F.n()\n",
    "print('\\nThe exact area by integration of a function is =',A_ex)\n",
    "\n",
    "DeltaA = abs(A_ex-SUM_A)\n",
    "print('\\nThe error made by integrating a fuction using... \\nSimpson rule of {} rectangles and exact solution is = {}\\n'.format(N,DeltaA))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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