right hand sum, left hand sum, and integration example in Python

integrals.ipynb

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      "source": "from functools import partial, singledispatch\nfrom itertools import islice, count\nfrom sympy import sympify, init_printing\nfrom sympy import integrate as _integrate\nfrom textwrap import dedent\nfrom mpmath import e\nimport sympy\n\ninit_printing()\n\ndef _sum(left_or_right, y_values, delta_x):\n    \"\"\"\n    Return the left or right hand sum of a list of y values.\n    \"\"\"\n    # make sure y_values is a list\n    y_values = list(y_values)\n    \n    if left_or_right == 'left':\n        y_values = y_values[:-1]\n    elif left_or_right == 'right':\n        y_values = y_values[1:]\n        \n    return sum(y*delta_x for y in y_values)\n\nlhs = partial(_sum, 'left')\nrhs = partial(_sum, 'right')\n\n@singledispatch\ndef integrate():\n    pass\n\n@integrate.register(dict)        \ndef _(observations, a_b=None, n=None, delta_x=None, verbose=False):\n    \"\"\"\n    Given a table of x, y values as a dictionary\n    return the definite integral.\n    \"\"\"\n    if delta_x is not None and n is not None:\n        raise ValueError('delta x and n are mutually exclusive')\n        \n    # pair down table of values if delta x is given\n    if delta_x is not None:\n        observations = {\n            k:v for k,v in observations.items() if k % delta_x == 0\n        }\n        x_vals = list(observations.keys())\n    else:\n        x_vals = list(observations.keys())\n        delta_x = x_vals[1]-x_vals[0]\n    \n    \n    \n    # n is the number of times we evenly split the observations\n    # by default, we'll choose the maximum amount of splits possible\n    \n    n = len(observations)-1 if n is None else n\n    \n    # let a and b be the largest and smallest values\n    # in our list of x values by default\n    \n    if a_b is not None:\n        a, b = a_b\n    else:\n        a = x_vals[0]\n        b = x_vals[-1]\n    \n    # create the list of y values that we will compute\n    # the right and left hand sums of\n    start = x_vals.index(a)\n    stop = x_vals.index(b)+1\n    step = len(observations) // n    \n    y_vals = list(observations.values())[start:stop:step]\n    \n    # return the average of the left and right hand sums\n    left_hand_sum = lhs(y_vals, delta_x)\n    right_hand_sum = rhs(y_vals, delta_x)\n    result = (left_hand_sum + right_hand_sum) / 2\n    \n    if verbose:\n        print(dedent(f\"\"\"\n        inputs: {observations}\n        \n        a             : {a} (x-axis lower limit)\n        b             : {b} (x-axis upper limit)\n        n             : {n} (the number of times we split the graph)\n        delta x       : {delta_x} (the interval between numbers on the x-axis)\n        left hand sum : {left_hand_sum}\n        right hand sum: {right_hand_sum}\n        \n        result: {result}\n        \"\"\"))\n    \n    return result\n\n@integrate.register(sympy.Basic)\ndef _(expr, a_b=None, n=None, verbose=False):\n    \"\"\"\n    Return the integral of a single-variable expression.\n    \"\"\"\n    \n    # ensure we replace e variable with constant e in\n    # expression\n    expr = expr.subs('e', e)\n    \n    # get symbol from expression\n    variable = list(expr.free_symbols)[0]\n    \n    # if n is given, create a table and use that to compute integral\n    if n is not None:\n        if a_b is None: raise ValueError('must specify range: a, b with n')\n        a, b = a_b\n        delta_x = (b-a)/n\n        x_vals = islice(count(a, delta_x), n+1)\n        observations = {\n            x: float(expr.subs(variable, x)) for x in x_vals\n        }\n        return integrate(observations, verbose=verbose)\n    \n    elif a_b is not None:\n        a, b = a_b\n        return _integrate(expr, (variable, a, b))\n    \n    return _integrate(expr)\n\n@integrate.register(str)\ndef _(expr, *args, **kwargs):\n    \"\"\"\n    Return the integral of a single-variable expression.\n    \"\"\"\n    return integrate(sympify(expr), *args, **kwargs)\n    \n    ",
      "execution_count": 9,
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      "cell_type": "code",
      "source": "observations = {20: 5, 22: 7, 24: 11, 26: 18, 28: 29, 30: 45}\n\nintegrate(observations, verbose=True) # -> 180 ",
      "execution_count": 2,
      "outputs": [
        {
          "output_type": "stream",
          "text": "\ninputs: {20: 5, 22: 7, 24: 11, 26: 18, 28: 29, 30: 45}\n\na             : 20 (x-axis lower limit)\nb             : 30 (x-axis upper limit)\nn             : 5 (the number of times we split the graph)\ndelta x       : 2 (the interval between numbers on the x-axis)\nleft hand sum : 140\nright hand sum: 220\n\nresult: 180.0\n\n",
          "name": "stdout"
        },
        {
          "output_type": "execute_result",
          "execution_count": 2,
          "data": {
            "text/plain": "180.0",
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAADsAAAASCAYAAADlhqZNAAAABHNCSVQICAgIfAhkiAAAAjpJREFU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\nHUdLYAAAAABJRU5ErkJggg==\n",
            "text/latex": "$$180.0$$"
          },
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      "cell_type": "code",
      "source": "integrate('2*x') # -> x**2\n\nintegrate('2*x', (0, 10)) # -> 100",
      "execution_count": 3,
      "outputs": [
        {
          "output_type": "execute_result",
          "execution_count": 3,
          "data": {
            "text/plain": " 2\nx ",
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAABcAAAAWCAYAAAArdgcFAAAABHNCSVQICAgIfAhkiAAAASdJREFU\nOI3t078rxVEYx/EXyiAWpRgsKJRffwCLTMrEZCKrf4DJxoIMks3KQlmwK3UHZZCJDXWHKwNFXMP3\n3Pq6vt873PtNBp86dZ7zPOf9POec5/CHtYQcnpHHMQaygp9iPgAHcYhHtGaVIK5mfGAK6jOGtwRm\nIWMuOMAlGrIGb+AeXWkBZyhiumy9DnvBt5awbxMP6KuUfVj0INe+H209gHcT9myJOqS/ErikUoVz\nwV4O9r6fDbAt6vFxtMdGcxq8E6+4w2IAn6AxIbaYMlYqVb8aCzxHU6XgNKX1eT42X8BLNfAkzeJT\n9PpF7GQFnsQbrtCGG7yjt1bwqOj4t+gIazOi6o9qAY/gSXQV3WW+XEgwVg24R/QJChhK8E8E+EU1\n8H/9rr4AkGRDHpkg5ysAAAAASUVORK5CYII=\n",
            "text/latex": "$$x^{2}$$"
          },
          "metadata": {}
        },
        {
          "output_type": "execute_result",
          "execution_count": 3,
          "data": {
            "text/plain": "100",
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAACcAAAASCAYAAADYFMcrAAAABHNCSVQICAgIfAhkiAAAASxJREFU\nSInt1bEuQ2EYxvFfjUQQiRiYmtYlEFM7WQ1uw+YCjCaRWEwsLoCRwSLMNlRCLURCgjIwYOjXRI6e\ntL7TRogn+fLmvM/7z3lOcvJ+/CLNYRX7eMQ7NlswY1jHFV5QxQqGOsw4CoFqOG4jXB43YW4LS9gL\nzycY7hADyiggh1Ib4XbCzHyivxz6ax1ivqhVuHzwL9CT8PrxhGf0xTLJge+oHOou3hJeDQfoxVQs\nkyXcRKiVFP8s1GIskyXcQKgPKX6jPxjLZAnXdWUJ1/jKgRS/0b+PZbKEOw21mOIXQv38f8UwTVXy\nw6skSzi6vIRziYHZcGAUMzhXv2vhFguf5vM4xAi21a+8SfV9VsE07hLviGHAYkifdqpNmHFs4Bqv\nuNT6Eo9h/vU39AGTVH07+0lGTwAAAABJRU5ErkJggg==\n",
            "text/latex": "$$100$$"
          },
          "metadata": {}
        }
      ]
    },
    {
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          "start_time": "2017-05-01T00:26:36.137182Z",
          "end_time": "2017-05-01T00:26:36.289456Z"
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        "trusted": true
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      "cell_type": "code",
      "source": "integrate('2**x', (0, 6), n=2, verbose=True)",
      "execution_count": 4,
      "outputs": [
        {
          "output_type": "stream",
          "text": "\ninputs: {0: 1.0, 3.0: 8.0, 6.0: 64.0}\n\na             : 0 (x-axis lower limit)\nb             : 6.0 (x-axis upper limit)\nn             : 2 (the number of times we split the graph)\ndelta x       : 3.0 (the interval between numbers on the x-axis)\nleft hand sum : 27.0\nright hand sum: 216.0\n\nresult: 121.5\n\n",
          "name": "stdout"
        },
        {
          "output_type": "execute_result",
          "execution_count": 4,
          "data": {
            "text/plain": "121.5",
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAADoAAAASCAYAAAAKRM1zAAAABHNCSVQICAgIfAhkiAAAAfZJREFU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\nRK5CYII=\n",
            "text/latex": "$$121.5$$"
          },
          "metadata": {}
        }
      ]
    },
    {
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        "trusted": true,
        "ExecuteTime": {
          "start_time": "2017-05-01T00:29:54.031338Z",
          "end_time": "2017-05-01T00:29:54.171287Z"
        }
      },
      "cell_type": "code",
      "source": "observations = {\n    0: 48,\n    2: 45,\n    4: 36,\n    6: 21,\n    8: 0,\n}\n\nintegrate(observations, delta_x=4, verbose=True)",
      "execution_count": 10,
      "outputs": [
        {
          "output_type": "stream",
          "text": "\ninputs: {0: 48, 4: 36, 8: 0}\n\na             : 0 (x-axis lower limit)\nb             : 8 (x-axis upper limit)\nn             : 2 (the number of times we split the graph)\ndelta x       : 4 (the interval between numbers on the x-axis)\nleft hand sum : 336\nright hand sum: 144\n\nresult: 240.0\n\n",
          "name": "stdout"
        },
        {
          "output_type": "execute_result",
          "execution_count": 10,
          "data": {
            "text/plain": "240.0",
            "image/png": "iVBORw0KGgoAAAANSUhEUgAAADsAAAASCAYAAADlhqZNAAAABHNCSVQICAgIfAhkiAAAAjhJREFU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\nxBYAAAAASUVORK5CYII=\n",
            "text/latex": "$$240.0$$"
          },
          "metadata": {}
        }
      ]
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