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  {
   "cells": [
    {
     "cell_type": "heading",
     "level": 1,
     "metadata": {},
     "source": [
      "Basic Numerical Integration: the Trapezoid Rule"
     ]
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "A simple illustration of the trapezoid rule for definite integration:\n",
      "\n",
      "$$\n",
      "\\int_{a}^{b} f(x)\\, dx \\approx \\frac{1}{2} \\sum_{k=1}^{N} \\left( x_{k} - x_{k-1} \\right) \\left( f(x_{k}) + f(x_{k-1}) \\right).\n",
      "$$\n",
      "<br>\n",
      "First, we define a simple function and sample it between 0 and 10 at 200 points"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "%matplotlib inline\n",
      "import numpy as np\n",
      "import matplotlib.pyplot as plt"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 1
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "def f(x):\n",
      "    return (x-3)*(x-5)*(x-7)+85\n",
      "\n",
      "x = np.linspace(0, 10, 200)\n",
      "y = f(x)"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 2
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Choose a region to integrate over and take only a few points in that region"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": true,
     "input": [
      "a, b = 1, 9\n",
      "xint = x[np.logical_and(x>=a, x<=b)][::30]\n",
      "yint = y[np.logical_and(x>=a, x<=b)][::30]"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [],
     "prompt_number": 3
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Plot both the function and the area below it in the trapezoid approximation"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "plt.plot(x, y, lw=2)\n",
      "plt.axis([0, 10, 0, 140])\n",
      "plt.fill_between(xint, 0, yint, facecolor='gray', alpha=0.4)\n",
      "plt.text(0.5 * (a + b), 30,r\"$\\int_a^b f(x)dx$\", horizontalalignment='center', fontsize=20);"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "metadata": {},
       "output_type": "display_data",
       "png": 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a5flQlpd6r/J8g9K81CnO0caybbrj0B6zaY+m7vnXHkv9d2gOQ7KfzbEiJ8xw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\nakRGhwMHDgCp3aPh8tJLL5GXl8e+fft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       "text": [
        "<matplotlib.figure.Figure at 0x1109fec90>"
       ]
      }
     ],
     "prompt_number": 4
    },
    {
     "cell_type": "markdown",
     "metadata": {},
     "source": [
      "Compute the integral both at high accuracy and with the trapezoid approximation"
     ]
    },
    {
     "cell_type": "code",
     "collapsed": false,
     "input": [
      "from __future__ import print_function\n",
      "from scipy.integrate import quad, trapz\n",
      "integral, error = quad(f, 1, 9)\n",
      "print(\"The integral is:\", integral, \"+/-\", error)\n",
      "print(\"The trapezoid approximation with\", len(xint), \"points is:\", trapz(yint, xint))"
     ],
     "language": "python",
     "metadata": {},
     "outputs": [
      {
       "output_type": "stream",
       "stream": "stdout",
       "text": [
        "The integral is: 680.0 +/- 7.54951656745e-12\n",
        "The trapezoid approximation with 6 points is: 621.286411141\n"
       ]
      }
     ],
     "prompt_number": 5
    }
   ],
   "metadata": {}
  }
 ]
}