05a_integral.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[^ gor: Uvod](00_uvod.ipynb)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Ploščina lika\n",
    "**Ploščino lika** med grafoma funkcij $f(x)$ in $g(x)$ na intervalu $[a,b]$ lahko izračunamo s preprosto formulo\n",
    "$$P = \\int_a^b\\left|f(x)-g(x)\\right | dx.$$\n",
    "Za bolj komplicirane like, lahko formulo na različnih intervalih kombiniramo.\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Primer\n",
    "Izračunaj skupno ploščino likov, ki jih omejujeta krivulji $y=x^2-x-2$ in $y=x^3-x^2+1$.  \n",
    "\n",
    "### Rešitev\n",
    "Najprej si narišemo graf, da si bomo vse skupaj lažje predstavljali"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 77,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "<matplotlib.text.Text at 0x7f059bd21cc0>"
      ]
     },
     "execution_count": 77,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
      "image/png": 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5nPvuvY9Gjf53K53L5WLM6NG8+txzXF5QwJsOB2dZGHJ1ywAmizAuIYGMmBge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\n6P8B07rifoY+XtsAAAAASUVORK5CYII=\n",
      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f059d66fb38>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "%matplotlib inline\n",
    "from matplotlib import pyplot as plt\n",
    "import numpy as np\n",
    "f = lambda x: x**2-x-2\n",
    "g = lambda x: x**3-2*x**2-3*x+1\n",
    "fig, ax = plt.subplots()\n",
    "x = np.linspace(-2,4)\n",
    "ax.fill_between(x, f(x),g(x),where = f(x)>=g(x), facecolor='green',interpolate=True)\n",
    "ax.fill_between(x, f(x), g(x), where = f(x)<= g(x),facecolor='red',interpolate=True)\n",
    "plt.title(\"Liki med dvema krivuljama.\")\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Krivulji omejujeta dva lika. Meje za integracijo izračunamo tako, da poiščemo presečišča. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 65,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAARoAAAAWBAMAAADtKE5GAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\n",
      "text/latex": [
       "$$x^{2} - x - 2 = x^{3} - 2 x^{2} - 3 x + 1$$"
      ],
      "text/plain": [
       " 2            3      2          \n",
       "x  - x - 2 = x  - 2⋅x  - 3⋅x + 1"
      ]
     },
     "execution_count": 65,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "import sympy as sym\n",
    "sym.init_printing()\n",
    "x = sym.Symbol('x')\n",
    "enacba = sym.Eq(f(x),g(x))\n",
    "enacba"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 66,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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DLxg2IjGhi9pWSUvy2TF3tRkfvRFLRcv6zuYZNSP3OX3ZWvJ4b/lNzy42d+sRi+DDpI7w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\ngg==\n",
      "text/latex": [
       "$$\\left [ 1 + \\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{1}{2} + \\frac{\\sqrt{1419} i}{18}} + \\frac{5}{3 \\left(- \\frac{1}{2} - \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{1}{2} + \\frac{\\sqrt{1419} i}{18}}}, \\quad 1 + \\frac{5}{3 \\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{1}{2} + \\frac{\\sqrt{1419} i}{18}}} + \\left(- \\frac{1}{2} + \\frac{\\sqrt{3} i}{2}\\right) \\sqrt[3]{\\frac{1}{2} + \\frac{\\sqrt{1419} i}{18}}, \\quad 1 + \\frac{5}{3 \\sqrt[3]{\\frac{1}{2} + \\frac{\\sqrt{1419} i}{18}}} + \\sqrt[3]{\\frac{1}{2} + \\frac{\\sqrt{1419} i}{18}}\\right ]$$"
      ],
      "text/plain": [
       "⎡                         ________________                                    \n",
       "⎢    ⎛        ___  ⎞     ╱       ______                                       \n",
       "⎢    ⎜  1   ╲╱ 3 ⋅ⅈ⎟    ╱  1   ╲╱ 1419 ⋅ⅈ                       5             \n",
       "⎢1 + ⎜- ─ - ───────⎟⋅3 ╱   ─ + ──────────  + ─────────────────────────────────\n",
       "⎢    ⎝  2      2   ⎠ ╲╱    2       18                               __________\n",
       "⎢                                              ⎛        ___  ⎞     ╱       ___\n",
       "⎢                                              ⎜  1   ╲╱ 3 ⋅ⅈ⎟    ╱  1   ╲╱ 14\n",
       "⎢                                            3⋅⎜- ─ - ───────⎟⋅3 ╱   ─ + ─────\n",
       "⎣                                              ⎝  2      2   ⎠ ╲╱    2       1\n",
       "\n",
       "                                                                           ___\n",
       "                                                      ⎛        ___  ⎞     ╱   \n",
       "                               5                      ⎜  1   ╲╱ 3 ⋅ⅈ⎟    ╱  1 \n",
       "──────, 1 + ─────────────────────────────────────── + ⎜- ─ + ───────⎟⋅3 ╱   ─ \n",
       "______                             ________________   ⎝  2      2   ⎠ ╲╱    2 \n",
       "___           ⎛        ___  ⎞     ╱       ______                              \n",
       "19 ⋅ⅈ         ⎜  1   ╲╱ 3 ⋅ⅈ⎟    ╱  1   ╲╱ 1419 ⋅ⅈ                            \n",
       "─────       3⋅⎜- ─ + ───────⎟⋅3 ╱   ─ + ──────────                            \n",
       "8             ⎝  2      2   ⎠ ╲╱    2       18                                \n",
       "\n",
       "_____________                                     ________________⎤\n",
       "    ______                                       ╱       ______   ⎥\n",
       "  ╲╱ 1419 ⋅ⅈ                  5                 ╱  1   ╲╱ 1419 ⋅ⅈ ⎥\n",
       "+ ────────── , 1 + ─────────────────────── + 3 ╱   ─ + ────────── ⎥\n",
       "      18                  ________________   ╲╱    2       18     ⎥\n",
       "                         ╱       ______                           ⎥\n",
       "                        ╱  1   ╲╱ 1419 ⋅ⅈ                         ⎥\n",
       "                   3⋅3 ╱   ─ + ──────────                         ⎥\n",
       "                     ╲╱    2       18                             ⎦"
      ]
     },
     "execution_count": 66,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sym.solve(enacba,x)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Formule za presečišča so precej grda, zato bomo raje uporabili kar približke s števili s plavajočo vejico (float)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 74,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAA7wAAAAbBAMAAACtnJ7PAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYOAoAAAAAElFTkSuQmCC\n",
      "text/latex": [
       "$$\\left [ 0.798360324276595 + 5.0 \\cdot 10^{-23} i, \\quad -1.12841906384458 - 5.0 \\cdot 10^{-23} i, \\quad 3.33005873956798 - 7.0 \\cdot 10^{-21} i\\right ]$$"
      ],
      "text/plain": [
       "[0.798360324276595 + 0.e-22⋅ⅈ, -1.12841906384458 - - -0.e-22⋅ⅈ, 3.330058739567\n",
       "98 - - -0.e-20⋅ⅈ]"
      ]
     },
     "execution_count": 74,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "resitve = [x.evalf() for x in sym.solve(enacba,x)]\n",
    "resitve"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Opazimo, da so rešitve kompleksne. Čudno! Ampak imaginarni del je zelo majhen. Očitno gre za zaokrožitveno napako, saj je na grafu lepo vindo, da so rešitve realne. Vzamemo le realni del."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 75,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAkEAAAAUBAMAAAB49vTzAAAAMFBMVEX///8AAAAAAAAAAAAAAAAA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\nYII=\n",
      "text/latex": [
       "$$\\left [ -1.12841906384458, \\quad 0.798360324276595, \\quad 3.33005873956798\\right ]$$"
      ],
      "text/plain": [
       "[-1.12841906384458, 0.798360324276595, 3.33005873956798]"
      ]
     },
     "execution_count": 75,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "resitve = [sym.re(x) for x in resitve]\n",
    "resitve.sort()\n",
    "resitve"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 78,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "image/png": 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Br3d60e3pbpf0lfWm5ORkFi1axNzp05k7cyYpCQm0A/6WkkJb4KpiicLJBmwC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\nqPHzz+zevfvyIGtByrMppJD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      "text/plain": [
       "<matplotlib.figure.Figure at 0x7f059d66fb38>"
      ]
     },
     "execution_count": 78,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "for i in range(len(resitve)):\n",
    "    xi = resitve[i]\n",
    "    ax.plot(xi,f(xi),'ko')\n",
    "    ax.annotate('$(x_%d,f(x_%d))$'%(i,i),xy = (xi,f(xi)),xytext=(xi,f(xi)-3))\n",
    "fig"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Ploščina\n",
    "\n",
    "Če uporabimo formulo za ploščino, dobimo integral z absolutno vrednostjo, ki ga razdelimo na dva dela\n",
    "$$\\int_{x_0}^{x_2}\\left|f(x)-g(x)\\right|dx = \\int_{x_0}^{x_1}\\left(g(x)-f(x)\\right)dx+\\int_{x_1}^{x_2}\\left(f(x)-g(x)\\right)dx$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 87,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Skupna ploščina obeh likov je približno enaka 12.801220\n"
     ]
    }
   ],
   "source": [
    "x = sym.Symbol('x')\n",
    "P = sym.integrate(g(x)-f(x),(x,resitve[0],resitve[1])) + sym.integrate(f(x)-g(x),(x,resitve[1],resitve[2]))\n",
    "print(\"Skupna ploščina obeh likov je približno enaka %f\" % P)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "[<< nazaj: odvod - ekstremi](04c_ekstremi.ipynb) [naprej: integral>>](05a_integral.pynb)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 55,
   "metadata": {
    "collapsed": false
   },
   "outputs": [
    {
     "data": {
      "text/html": [
       "\n",
       "<div id=\"disqus_thread\"></div>\n",
       "\n",
       "<script type=\"text/javascript\">\n",
       "    var disqus_shortname = \"matpy\";\n",
       "    \n",
       "</script>\n",
       "<script src=\"https://matpy.disqus.com/embed.js\" async=\"true\">\n",
       "</script>\n",
       "\n",
       "<noscript>\n",
       "    Please enable JavaScript to view the\n",
       "    <a href=\"https://disqus.com/?ref_noscript\" rel=\"nofollow\">\n",
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       "<IPython.core.display.HTML object>"
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     "metadata": {},
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   ],
   "source": [
    "import disqus\n",
    "%reload_ext disqus\n",
    "%disqus matpy"
   ]
  }
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