1{
2 "cells": [
3  {
4   "cell_type": "markdown",
5   "metadata": {},
6   "source": [
7    "# Basic Numerical Integration: the Trapezoid Rule"
8   ]
9  },
10  {
11   "cell_type": "markdown",
12   "metadata": {},
13   "source": [
14    "A simple illustration of the trapezoid rule for definite integration:\n",
15    "\n",
16    "$$\n",
17    "\\int_{a}^{b} f(x)\\, dx \\approx \\frac{1}{2} \\sum_{k=1}^{N} \\left( x_{k} - x_{k-1} \\right) \\left( f(x_{k}) + f(x_{k-1}) \\right).\n",
18    "$$\n",
19    "<br>\n",
20    "First, we define a simple function and sample it between 0 and 10 at 200 points"
21   ]
22  },
23  {
24   "cell_type": "code",
25   "execution_count": 1,
26   "metadata": {
27    "collapsed": false
28   },
29   "outputs": [],
30   "source": [
31    "%matplotlib inline\n",
32    "import numpy as np\n",
33    "import matplotlib.pyplot as plt"
34   ]
35  },
36  {
37   "cell_type": "code",
38   "execution_count": 2,
39   "metadata": {
40    "collapsed": false
41   },
42   "outputs": [],
43   "source": [
44    "def f(x):\n",
45    "    return (x-3)*(x-5)*(x-7)+85\n",
46    "\n",
47    "x = np.linspace(0, 10, 200)\n",
48    "y = f(x)"
49   ]
50  },
51  {
52   "cell_type": "markdown",
53   "metadata": {},
54   "source": [
55    "Choose a region to integrate over and take only a few points in that region"
56   ]
57  },
58  {
59   "cell_type": "code",
60   "execution_count": 3,
61   "metadata": {
62    "collapsed": true
63   },
64   "outputs": [],
65   "source": [
66    "a, b = 1, 8 # the left and right boundaries\n",
67    "N = 5 # the number of points\n",
68    "xint = np.linspace(a, b, N)\n",
69    "yint = f(xint)"
70   ]
71  },
72  {
73   "cell_type": "markdown",
74   "metadata": {},
75   "source": [
76    "Plot both the function and the area below it in the trapezoid approximation"
77   ]
78  },
79  {
80   "cell_type": "code",
81   "execution_count": 4,
82   "metadata": {
83    "collapsed": false
84   },
85   "outputs": [
86    {
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895       "AAAAAACw5v8DPWODrRZIv60AAAAASUVORK5CYII=\n"
896      ],
897      "text/plain": [
898       "<matplotlib.figure.Figure at 0x10664cfd0>"
899      ]
900     },
901     "metadata": {
902      "image/png": {
903       "height": 381,
904       "width": 595
905      }
906     },
907     "output_type": "display_data"
908    }
909   ],
910   "source": [
911    "plt.plot(x, y, lw=2)\n",
912    "plt.axis([0, 9, 0, 140])\n",
913    "plt.fill_between(xint, 0, yint, facecolor='gray', alpha=0.4)\n",
914    "plt.text(0.5 * (a + b), 30,r\"$\\int_a^b f(x)dx$\", horizontalalignment='center', fontsize=20);"
915   ]
916  },
917  {
918   "cell_type": "markdown",
919   "metadata": {},
920   "source": [
921    "Compute the integral both at high accuracy and with the trapezoid approximation"
922   ]
923  },
924  {
925   "cell_type": "code",
926   "execution_count": 5,
927   "metadata": {
928    "collapsed": false
929   },
930   "outputs": [
931    {
932     "name": "stdout",
933     "output_type": "stream",
934     "text": [
935      "The integral is: 565.2499999999999 +/- 6.275535646693696e-12\n",
936      "The trapezoid approximation with 5 points is: 559.890625\n"
937     ]
938    }
939   ],
940   "source": [
941    "from __future__ import print_function\n",
942    "from scipy.integrate import quad\n",
943    "integral, error = quad(f, a, b)\n",
944    "integral_trapezoid = sum( (xint[1:] - xint[:-1]) * (yint[1:] + yint[:-1]) ) / 2\n",
945    "print(\"The integral is:\", integral, \"+/-\", error)\n",
946    "print(\"The trapezoid approximation with\", len(xint), \"points is:\", integral_trapezoid)"
947   ]
948  }
949 ],
950 "metadata": {
951  "kernelspec": {
952   "display_name": "Python 3",
953   "language": "python",
954   "name": "python3"
955  },
956  "language_info": {
957   "codemirror_mode": {
958    "name": "ipython",
959    "version": 3
960   },
961   "file_extension": ".py",
962   "mimetype": "text/x-python",
963   "name": "python",
964   "nbconvert_exporter": "python",
965   "pygments_lexer": "ipython3",
966   "version": "3.4.2"
967  }
968 },
969 "nbformat": 4,
970 "nbformat_minor": 0
971}
972