{
    "worksheets": [
        {
            "cells": [
                {
                    "source": "Basic numerical integration: the trapezoid rule\n===============================================\n\nA 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>\nFirst, we define a simple function and sample it between 0 and 10 at 200 points", 
                    "cell_type": "markdown"
                }, 
                {
                    "cell_type": "code", 
                    "language": "python", 
                    "outputs": [], 
                    "collapsed": true, 
                    "prompt_number": 1, 
                    "input": "def f(x):\n    return (x-3)*(x-5)*(x-7)+85\n\nx = linspace(0, 10, 200)\ny = f(x)"
                }, 
                {
                    "source": "Choose a region to integrate over and take only a few points in that region", 
                    "cell_type": "markdown"
                }, 
                {
                    "cell_type": "code", 
                    "language": "python", 
                    "outputs": [], 
                    "collapsed": true, 
                    "prompt_number": 2, 
                    "input": "a, b = 1, 9\nxint = x[logical_and(x>=a, x<=b)][::30]\nyint = y[logical_and(x>=a, x<=b)][::30]"
                }, 
                {
                    "source": "Plot both the function and the area below it in the trapezoid approximation", 
                    "cell_type": "markdown"
                }, 
                {
                    "cell_type": "code", 
                    "language": "python", 
                    "outputs": [
                        {
                            "output_type": "display_data", 
                            "png": 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Nhg0bMDMzo2/fvuoWKkQHVlVVxS+//MKCBQvULuWONDc79eqW8fT0xNnZmcTE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\nWmFhH6GH2icvZSFoIVpOuwr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                        }
                    ], 
                    "collapsed": false, 
                    "prompt_number": 3, 
                    "input": "plot(x, y, lw=2)\naxis([0, 10, 0, 140])\nfill_between(xint, 0, yint, facecolor='gray', alpha=0.4)\ntext(0.5 * (a + b), 30,r\"$\\int_a^b f(x)dx$\", horizontalalignment='center', fontsize=20);"
                }, 
                {
                    "source": "Compute the integral both at high accuracy and with the trapezoid approximation", 
                    "cell_type": "markdown"
                }, 
                {
                    "cell_type": "code", 
                    "language": "python", 
                    "outputs": [
                        {
                            "output_type": "stream", 
                            "text": "The ingtegral is: 680.0 +/- 7.54951656745e-12\nThe trapezoid approximation with 6 points is: 621.286411141"
                        }
                    ], 
                    "collapsed": false, 
                    "prompt_number": 4, 
                    "input": "from scipy.integrate import quad, trapz\nintegral, error = quad(f, 1, 9)\nprint \"The ingtegral is:\", integral, \"+/-\", error\nprint \"The trapezoid approximation with\", len(xint), \"points is:\", trapz(yint, xint)"
                }, 
                {
                    "input": "", 
                    "cell_type": "code", 
                    "collapsed": true, 
                    "language": "python", 
                    "outputs": []
                }
            ]
        }
    ], 
    "metadata": {
        "name": "trapezoid rule integration"
    }, 
    "nbformat": 2
}