integration_practice.ipynb

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                "# Non-linear Pendulm\n",
                "\n",
                "The pendulum is a classic physics problem that most students will see in their introductory physics classes. \n",
                "\u003ccenter\u003e\n",
                "\n",
                "![Pendulm Force Diagram](https://upload.wikimedia.org/wikipedia/commons/6/66/Pendulum_gravity.svg)\n",
                "\n",
                "\u003c/center\u003e\n",
                ""
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                "Deriving the equation of motion for the pendulum is a strait-forward application of Newton's Laws:\n",
                "\n",
                "$$F = ma$$\n",
                "\n",
                "where\n",
                "\n",
                "$$a = -gsin\\theta$$\n",
                "\n",
                "The arc length (distance the bob travels) is:\n",
                "\n",
                "$$s = l\\theta$$\n",
                "\n",
                "which allows us to write the velocity as:\n",
                "\n",
                "$$v = \\frac{ds}{dt} = l \\frac{d\\theta}{dt}$$\n",
                "\n",
                "and allows us to write the accelration as:\n",
                "\n",
                "$$a = \\frac{d^2s}{dt^2} = l \\frac{d^2\\theta}{dt^2}$$\n",
                "\n",
                "Combining all of this information: \n",
                "\n",
                "$$l \\frac{d^2\\theta}{dt^2} = -gsin\\theta$$\n",
                "\n",
                "$$\\frac{d^2\\theta}{dt^2} + \\frac {g}{l} sin\\theta = 0 $$"
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                "## Small Angle Approx.\n",
                "\n",
                "From here, if we were just trying to keep things simple, we would apply the small angle approximation ($\\theta \\ll 1$), allowing us to say $sin\\theta \\approx \\theta$, producing the well known results:\n",
                "\n",
                "$$\\theta = \\theta_0 cos(\\sqrt{\\frac{g}{l}}t)$$\n",
                "\n",
                "and\n",
                "\n",
                "$$T_0 = 2\\pi\\sqrt{\\frac{l}{g}}$$"
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                "## Without Small Angle Approx.\n",
                "\n",
                "But, because this isn't introductory physics, let's investigate the pendulum's periodic motion without the small angle approximation. \n",
                "\n",
                "Using potential and kinetic energy, we can write:\n",
                "\n",
                "$$\\frac{1}{2}mv^2 = mgh$$\n",
                "\n",
                "where $h$ is the height our pendulum is at. So, the velocity is then:\n",
                "\n",
                "$$v = \\sqrt{2gh}$$\n",
                "\n",
                "Using the arc length formula from above, we can say:\n",
                "\n",
                "$$v = l\\frac{d\\theta}{dt} = \\sqrt{2gh}$$\n",
                "\n",
                "$$\\frac{d\\theta}{dt} = \\frac{\\sqrt{2gh}}{l}$$\n",
                "\n",
                "The initial height $h_0$ of our pendulum can be written as $lcos\\theta_0$, then some time later, the height will be $lcos\\theta$.\n",
                "\n",
                "The change in height is then:\n",
                "\n",
                "$$\\Delta h = l(cos\\theta - cos\\theta_0)$$\n",
                "\n",
                "So, our $\\frac{d\\theta}{dt}$ can now be written as:\n",
                "\n",
                "$$\\frac{d\\theta}{dt} = \\sqrt{\\frac{2g}{l}(cos\\theta - cos\\theta_0)}$$\n",
                "\n",
                "From here, we can invert this equation and multiply both sides by $d\\theta$:\n",
                "\n",
                "$$dt = \\frac{d\\theta}{\\sqrt{\\frac{2g}{l}(cos\\theta - cos\\theta_0)}}$$\n",
                "\n",
                "$$dt = \\sqrt{\\frac{l}{2g}}\\frac{d\\theta}{\\sqrt{(cos\\theta - cos\\theta_0)}}$$\n",
                "\n",
                "Finally, to get the period, we just need to integrate (note that if pick our $t_0$ to be when the pendulum is at its max height, then we can simply multipy by 4 and integrate from $\\theta = 0$ to $\\theta = \\theta_0$):\n",
                "\n",
                "$$T = 4\\sqrt{\\frac{l}{2g}}\\ \\int_{0}^{\\theta_0} \\frac{d\\theta}{\\sqrt{(cos\\theta - cos\\theta_0)}}$$\n",
                "\n",
                "If you try to solve this equation using our numerical techniques in python, you will run into errors and it will only return `nan` values. Instead, we need to apply some trigonometric functions. However, to save some time, we will simply write the final, usable form. \n",
                "\n",
                "$$T = 4\\sqrt{\\frac{l}{g}}\\ \\int_{0}^{\\frac{\\pi}{2}} \\frac{d\\phi}{\\sqrt{(1 - A^2sin^2\\phi)}}$$\n",
                "\n",
                "where:\n",
                "\n",
                "$$A = sin(\\frac{\\theta_{0}}{2})$$\n",
                "\n",
                "Note: this is an *eliptical integral*, in case you want to learn more.\n",
                ""
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                "### Our Integration Techniques\n",
                "Part 1.\n",
                "\n",
                "Using the integration techniques from Lab 4, find the period of a pendulum with mass $m = 1$kg and length $l = 1$m for all initial angles between 0 and 90 degrees. Use $N = 10, 100, 1000$, is there much difference between the integration methods?\n",
                "\n",
                "Note: for $\\theta_0 = 45\\degree$, you should get $T \\approx 2.08732$ seconds.\n",
                ""
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                        "text/plain": "      N  Trapezoidal Method     Simpson's Rule       Gauss Method\n0    10  2.0873200174795943 1.9480017374375171 2.0873200174796036\n1   100  2.0873200174795938 2.0733881894753865 2.0873200174795938\n2  1000  2.0873200174795943 2.0859268346791739 2.0873200174795938"
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                "import numpy as np\n",
                "import matplotlib.pyplot as plt\n",
                "import pandas as pd\n",
                "#############################################################################################\n",
                "#                          Using the integral teciniques from Lab 4                         #\n",
                "#############################################################################################\n",
                "def gauss(f,a,b,N):\n",
                "    y,wprime = np.polynomial.legendre.leggauss(N)\n",
                "    x = 0.5*(y + 1)*(b - a) + a\n",
                "    w = wprime* 0.5*(b-a)\n",
                "    return (f(x)*w).sum()\n",
                "\n",
                "def qsimp(f,a,b,N):\n",
                "    \"\"\"qsimp uses Simpson's rule to evaluate an approximation to the integral of f(x) on the interval [a, b] using N points.\n",
                "    \n",
                "    inputs\n",
                "    ------\n",
                "    f: function that can operate on numpy arrays\n",
                "    a: float representing the lower limit of the integral\n",
                "    b: float representing the upper limit of the integral\n",
                "    N: int representing the number of samples\n",
                "    \n",
                "    \"\"\"\n",
                "    if N%2!=1 and N\u003e=3:\n",
                "        raise ValueError(\"Simpson's rule requires that N, the number of points be odd and at least 3.\")\n",
                "    h = (b-a)/(N-1) # first we define the width between each x_i\n",
                "    x = a+np.arange(N)*h # np.arange(N) returns a np array; we can multiply it and add numbers to each element in it\n",
                "    \n",
                "    w = np.zeros(N) #initializing a weight array\n",
                "\n",
                "    for i in range(len(x)):\n",
                "        if i%2==0:\n",
                "            w[i] = h *4/3 # i is even\n",
                "        else:\n",
                "            w[i] = h *2/3 # i is odd\n",
                "    w[0] = w[-1] = h/3 # fixing end points\n",
                "        \n",
                "    data = f(x) # f is a function that we \"pass\" into the qtrap; here we have it evaluate at every point in x, \n",
                "                # storing the results in a new numpy array I called \"data\"\n",
                "    \n",
                "    # this is the actual integral: data*w is a new numpy array \n",
                "    # of the same length as data and w, which I call \"integrand\"\n",
                "    integrand = data*w\n",
                "    \n",
                "    # numpy arrays can sum themeselves by calling their .sum() method:\n",
                "    integral = integrand.sum()\n",
                "\n",
                "    return integral\n",
                "\n",
                "def qtrap(f,a,b,N):\n",
                "    \"\"\"qtrap evaluates the trapezoidal rule approximation to the integral of f(x) on the interval [a, b] using N points.\n",
                "    \n",
                "    inputs\n",
                "    ------\n",
                "    f: function that can operate on numpy arrays\n",
                "    a: float representing the lower limit of the integral\n",
                "    b: float representing the upper limit of the integral\n",
                "    N: int representing the number of samples\n",
                "    \n",
                "    \"\"\"\n",
                "    h = (b-a)/(N-1) # first we define h.\n",
                "    x = a+np.arange(N)*h # np.arange(N) returns a np array; we can multiply it and add numbers to each element in it\n",
                "    \n",
                "    w = h*np.ones(N) # this is the same trick as for x; multipying a bunch of ones from np.ones(N) by h\n",
                "    # since w is 1 on all interior points, and 1/2 on the first and last, we can just do\n",
                "    w[0] *= 0.5\n",
                "    w[-1] *= 0.5\n",
                "\n",
                "        \n",
                "    data = f(x) # f is a function that we \"pass\" into the qtrap; here we have it evaluate at every point in x, \n",
                "                # storing the results in a new numpy array I called \"data\"\n",
                "    \n",
                "    # this is the actual integral: data*w is a new numpy array \n",
                "    # of the same length as data and w, which I call \"integrand\"\n",
                "    integrand = data*w\n",
                "    \n",
                "    # numpy arrays can sum themeselves by calling their .sum() method:\n",
                "    integral = integrand.sum()\n",
                "    \n",
                "    return integral\n",
                "#############################################################################################\n",
                "m = 1\n",
                "l = 1\n",
                "g = 9.8\n",
                "theta_0 = np.pi/4\n",
                "\n",
                "integral_func = lambda phi: (1-(np.sin(theta_0/2))**2*(np.sin(phi))**2)**(-.5)\n",
                "\n",
                "T= lambda int_func_eval: 4 * (l/g)**(.5) * int_func_eval\n",
                "\n",
                "data_to_display = []\n",
                "N_ls = [10,100,1000]\n",
                "\n",
                "for N in N_ls:\n",
                "    gauss_eval = T(gauss(integral_func,0, np.pi/2,N))\n",
                "    qtrap_eval = T(qtrap(integral_func,0, np.pi/2,N))\n",
                "    qsimp_eval = T(qsimp(integral_func,0, np.pi/2,N+1)) # we use N+1 to make it odd-\u003erequired by Simpson's Law\n",
                "    data_to_display.append([N, qtrap_eval, qsimp_eval, gauss_eval])\n",
                "\n",
                "df = pd.DataFrame(data_to_display, columns = ['N', 'Trapezoidal Method', \"Simpson's Rule\", \"Gauss Method\"])\n",
                "\n",
                "pd.options.display.float_format = \"{:,.16f}\".format\n",
                "\n",
                "display(df)"
            ]
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                "Part 2.\n",
                "\n",
                "Compare our results to the small angle approx. result: $T_0 = 2\\pi\\sqrt{\\frac{l}{g}}$. Produce a plot showing Percent Error vs $\\theta_0$"
            ]
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\n",
                        "text/plain": "\u003cFigure size 640x480 with 1 Axes\u003e"
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "# The results when N is large enough are basically the same, I will use a large N and gauss's method\n",
                "# to obtain a sufficiently 'true' value\n",
                "T_0 = 2 * np.pi * (l/g)**(.5) # constant\n",
                "\n",
                "theta_0_ls = np.linspace(0,np.pi/2, 50)\n",
                "perc_err = []\n",
                "\n",
                "for theta_0 in theta_0_ls:\n",
                "    true = T(gauss(integral_func,0, np.pi/2,N)) # should still grab the correct theta_0\n",
                "    perc_err.append(abs(T_0 - true)/true * 100)\n",
                "\n",
                "\n",
                "plt.plot(theta_0_ls, perc_err)\n",
                "plt.xlabel(r\"$ \\theta_0 $\")\n",
                "plt.ylabel(\"Percent Error\")\n",
                "plt.title(\"The Percent Error in the Solution Using the Small Angle Approzimation\")\n",
                "plt.show()\n",
                ""
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "### Scipy Integration\n",
                "\n",
                "If you happen to need to integrate in the future, there is a decent chance you will simply want to use one of the pre-build functions in Python to do so. \n",
                "\n",
                "The Scipy Library offers many great pre-built functions ready to use, just make sure you are using them correctly. It is very easy to apply a pre-build function where it is not meant to be used if you don't fully understand what the function is doing. \n",
                "\n",
                "Here is a link to the [scipy.integrate](https://docs.scipy.org/doc/scipy/tutorial/integrate.html) documentation page. It is worth reviewing all that is there. \n",
                "\n",
                "Here, we will only look at 2 integration techniques from Scipy: `quad` and `fixed_quad`."
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 3,
            "metadata": {},
            "outputs": [
                {
                    "data": {
                        "text/html": "\u003cdiv\u003e\n\u003cstyle scoped\u003e\n    .dataframe tbody tr th:only-of-type {\n        vertical-align: middle;\n    }\n\n    .dataframe tbody tr th {\n        vertical-align: top;\n    }\n\n    .dataframe thead th {\n        text-align: right;\n    }\n\u003c/style\u003e\n\u003ctable border=\"1\" class=\"dataframe\"\u003e\n  \u003cthead\u003e\n    \u003ctr style=\"text-align: right;\"\u003e\n      \u003cth\u003e\u003c/th\u003e\n      \u003cth\u003eIntegral Result\u003c/th\u003e\n      \u003cth\u003eIntegral Error\u003c/th\u003e\n    \u003c/tr\u003e\n  \u003c/thead\u003e\n  \u003ctbody\u003e\n    \u003ctr\u003e\n      \u003cth\u003equad\u003c/th\u003e\n      \u003ctd\u003e2.0000000000000000\u003c/td\u003e\n      \u003ctd\u003e0.0000000000000222\u003c/td\u003e\n    \u003c/tr\u003e\n    \u003ctr\u003e\n      \u003cth\u003egauss\u003c/th\u003e\n      \u003ctd\u003e2.0000001102844727\u003c/td\u003e\n      \u003ctd\u003eNaN\u003c/td\u003e\n    \u003c/tr\u003e\n  \u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e",
                        "text/plain": "         Integral Result     Integral Error\nquad  2.0000000000000000 0.0000000000000222\ngauss 2.0000001102844727                NaN"
                    },
                    "execution_count": 3,
                    "metadata": {},
                    "output_type": "execute_result"
                }
            ],
            "source": [
                "import scipy.integrate as integrate\n",
                "import numpy as np\n",
                "import pandas as pd\n",
                "\n",
                "start = 0\n",
                "stop = np.pi\n",
                "\n",
                "# my array for the results to go in\n",
                "I = np.zeros((2,2)) \n",
                "\n",
                "# start with regular quad\n",
                "[I[0,0], I[0,1]] = (integrate.quad(np.sin, start, stop))\n",
                "\n",
                "# now with gauss\n",
                "[I[1,0], I[1,1]] = (integrate.fixed_quad(np.sin, start, stop))\n",
                "\n",
                "df = pd.DataFrame(I, columns =['Integral Result', 'Integral Error'], index=['quad','gauss'])\n",
                "pd.options.display.float_format = \"{:,.16f}\".format\n",
                "df"
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "When you use the function there are two outputs: the integral estimate and an estimate of the error. The documentation on the [scipy.integrate](https://docs.scipy.org/doc/scipy/tutorial/integrate.html#general-integration-quad) website doesn't go into as much detail as I'd like, but you can find a [stackoverflow](https://stackoverflow.com/questions/41662492/what-is-the-meaning-of-error-obtained-in-dblquad-integration-using-scipy) discussion about how the error is calculated. \n",
                "\n",
                "The important thing to take form this is while there is an error estimate, that does not mean the true result of the integral is simply +/- the estimated result. It is very possible scipy does not produce a reliable value, so make sure your results make sense! \n",
                "\n",
                ""
            ]
        },
        {
            "attachments": {},
            "cell_type": "markdown",
            "metadata": {},
            "source": [
                "Using both `integrate.quad` and `integrate.fixed_quad`, calculate the period of the pendulum with mass $m = 1$kg and length $l = 1$m for all initial angles between 0 and 90 degrees. How do your results from scipy compare with our integration techniques from lab 4? "
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 6,
            "metadata": {},
            "outputs": [
                {
                    "name": "stdout",
                    "output_type": "stream",
                    "text": "Comparing the Results of Different Integration Methods using N=1000 for Ours:\n"
                },
                {
                    "data": {
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\u003ctd\u003e2.2989609594728613\u003c/td\u003e\n      \u003ctd\u003e2.2989656950360238\u003c/td\u003e\n    \u003c/tr\u003e\n    \u003ctr\u003e\n      \u003cth\u003e83\u003c/th\u003e\n      \u003ctd\u003e83\u003c/td\u003e\n      \u003ctd\u003e2.3071479452021597\u003c/td\u003e\n      \u003ctd\u003e2.3055856314783973\u003c/td\u003e\n      \u003ctd\u003e2.3071479452021562\u003c/td\u003e\n      \u003ctd\u003e2.3071479452021602\u003c/td\u003e\n      \u003ctd\u003e2.3071531593086028\u003c/td\u003e\n    \u003c/tr\u003e\n    \u003ctr\u003e\n      \u003cth\u003e84\u003c/th\u003e\n      \u003ctd\u003e84\u003c/td\u003e\n      \u003ctd\u003e2.3154929250769052\u003c/td\u003e\n      \u003ctd\u003e2.3139236264719010\u003c/td\u003e\n      \u003ctd\u003e2.3154929250769012\u003c/td\u003e\n      \u003ctd\u003e2.3154929250769056\u003c/td\u003e\n      \u003ctd\u003e2.3154986498747094\u003c/td\u003e\n    \u003c/tr\u003e\n    \u003ctr\u003e\n      \u003cth\u003e85\u003c/th\u003e\n      \u003ctd\u003e85\u003c/td\u003e\n      \u003ctd\u003e2.3239987930339860\u003c/td\u003e\n      \u003ctd\u003e2.3224223298035089\u003c/td\u003e\n      \u003ctd\u003e2.3239987930339816\u003c/td\u003e\n      \u003ctd\u003e2.3239987930339860\u003c/td\u003e\n      \u003ctd\u003e2.3240050606458675\u003c/td\u003e\n    \u003c/tr\u003e\n    \u003ctr\u003e\n      \u003cth\u003e86\u003c/th\u003e\n      \u003ctd\u003e86\u003c/td\u003e\n      \u003ctd\u003e2.3326685394768525\u003c/td\u003e\n      \u003ctd\u003e2.3310847264461940\u003c/td\u003e\n      \u003ctd\u003e2.3326685394768485\u003c/td\u003e\n      \u003ctd\u003e2.3326685394768529\u003c/td\u003e\n      \u003ctd\u003e2.3326753815594170\u003c/td\u003e\n    \u003c/tr\u003e\n    \u003ctr\u003e\n      \u003cth\u003e87\u003c/th\u003e\n      \u003ctd\u003e87\u003c/td\u003e\n      \u003ctd\u003e2.3415052552323639\u003c/td\u003e\n      \u003ctd\u003e2.3399139015571144\u003c/td\u003e\n      \u003ctd\u003e2.3415052552323599\u003c/td\u003e\n      \u003ctd\u003e2.3415052552323639\u003c/td\u003e\n      \u003ctd\u003e2.3415127024460078\u003c/td\u003e\n    \u003c/tr\u003e\n    \u003ctr\u003e\n      \u003cth\u003e88\u003c/th\u003e\n      \u003ctd\u003e88\u003c/td\u003e\n      \u003ctd\u003e2.3505121357210288\u003c/td\u003e\n      \u003ctd\u003e2.3489130446349558\u003c/td\u003e\n      \u003ctd\u003e2.3505121357210239\u003c/td\u003e\n      \u003ctd\u003e2.3505121357210288\u003c/td\u003e\n      \u003ctd\u003e2.3505202170981216\u003c/td\u003e\n    \u003c/tr\u003e\n    \u003ctr\u003e\n      \u003cth\u003e89\u003c/th\u003e\n      \u003ctd\u003e89\u003c/td\u003e\n      \u003ctd\u003e2.3596924853545147\u003c/td\u003e\n      \u003ctd\u003e2.3580854539036813\u003c/td\u003e\n      \u003ctd\u003e2.3596924853545094\u003c/td\u003e\n      \u003ctd\u003e2.3596924853545151\u003c/td\u003e\n      \u003ctd\u003e2.3597012275521689\u003c/td\u003e\n    \u003c/tr\u003e\n    \u003ctr\u003e\n      \u003cth\u003e90\u003c/th\u003e\n      \u003ctd\u003e90\u003c/td\u003e\n      \u003ctd\u003e2.3690497221753444\u003c/td\u003e\n      \u003ctd\u003e2.3674345409375510\u003c/td\u003e\n      \u003ctd\u003e2.3690497221753386\u003c/td\u003e\n      \u003ctd\u003e2.3690497221753448\u003c/td\u003e\n      \u003ctd\u003e2.3690591485976480\u003c/td\u003e\n    \u003c/tr\u003e\n  \u003c/tbody\u003e\n\u003c/table\u003e\n\u003c/div\u003e",
                        "text/plain": "    theta_0 (degrees)  Trapezoidal Method     Simpson's Rule  \\\n0                   0  2.0070899231544925 2.0057518632057239   \n1                   1  2.0071281359809907 2.0057900505566399   \n2                   2  2.0072427824638446 2.0059046206030477   \n3                   3  2.0074338866179295 2.0060955973307086   \n4                   4  2.0077014884840221 2.0063630207318464   \n5                   5  2.0080456441528001 2.0067069468291008   \n6                   6  2.0084664257984866 2.0071274477091099   \n7                   7  2.0089639217221720 2.0076246115657495   \n8                   8  2.0095382364048837 2.0081985427531039   \n9                   9  2.0101894905704585 2.0088493618482177   \n10                 10  2.0109178212583023 2.0095772057237196   \n11                 11  2.0117233819061293 2.0103822276304033   \n12                 12  2.0126063424427847 2.0112645972898804   \n13                 13  2.0135668893912779 2.0122245009974065   \n14                 14  2.0146052259821419 2.0132621417350380   \n15                 15  2.0157215722772830 2.0143777392952376   \n16                 16  2.0169161653044680 2.0155715304151087   \n17                 17  2.0181892592026331 2.0168437689214227   \n18                 18  2.0195411253781907 2.0181947258866346   \n19                 19  2.0209720526725592 2.0196246897960806   \n20                 20  2.0224823475411142 2.0211339667265906   \n21                 21  2.0240723342438121 2.0227228805367439   \n22                 22  2.0257423550477349 2.0243917730690191   \n23                 23  2.0274927704418282 2.0261410043641193   \n24                 24  2.0293239593641221 2.0279709528877428   \n25                 25  2.0312363194417378 2.0298820157701281   \n26                 26  2.0332302672440128 2.0318746090586806   \n27                 27  2.0353062385490932 2.0339491679840340   \n28                 28  2.0374646886243548 2.0361061472399253   \n29                 29  2.0397060925210457 2.0383460212772495   \n30                 30  2.0420309453835723 2.0406692846127270   \n31                 31  2.0444397627738518 2.0430764521526106   \n32                 32  2.0469330810112085 2.0455680595318970   \n33                 33  2.0495114575282884 2.0481446634695248   \n34                 34  2.0521754712435230 2.0508068421400814   \n35                 35  2.0549257229506743 2.0535551955625553   \n36                 36  2.0577628357260487 2.0563903460067157   \n37                 37  2.0606874553539765 2.0593129384177180   \n38                 38  2.0637002507712050 2.0623236408595771   \n39                 39  2.0668019145308785 2.0654231449781850   \n40                 40  2.0699931632868207 2.0686121664845758   \n41                 41  2.0732747382988665 2.0718914456592041   \n42                 42  2.0766474059600455 2.0752617478780055   \n43                 43  2.0801119583464427 2.0787238641610979   \n44                 44  2.0836692137906305 2.0822786117449792   \n45                 45  2.0873200174795943 2.0859268346791739   \n46                 46  2.0910652420781370 2.0896694044482840   \n47                 47  2.0949057883788029 2.0935072206205025   \n48                 48  2.0988425859794071 2.0974412115236531   \n49                 49  2.1028765939893264 2.1014723349499289   \n50                 50  2.1070088017657782 2.1056015788905333   \n51                 51  2.1112402296813602 2.1098299623015082   \n52                 52  2.1155719299242217 2.1141585359021007   \n53                 53  2.1200049873322935 2.1185883830071073   \n54                 54  2.1245405202630976 2.1231206203946971   \n55                 55  2.1291796815007356 2.1277563992113175   \n56                 56  2.1339236592017587 2.1324969059153727   \n57                 57  2.1387736778816957 2.1373433632614560   \n58                 58  2.1437309994441573 2.1422970313270309   \n59                 59  2.1487969242545013 2.1473592085835609   \n60                 60  2.1539727922602019 2.1525312330142032   \n61                 61  2.1592599841601636 2.1578144832803123   \n62                 62  2.1646599226253662 2.1632103799391262   \n63                 63  2.1701740735733712 2.1687203867151634   \n64                 64  2.1758039474993689 2.1743460118279878   \n65                 65  2.1815511008666135 2.1800888093791961   \n66                 66  2.1874171375592568 2.1859503808016090   \n67                 67  2.1934037104007968 2.1919323763738907   \n68                 68  2.1995125227415415 2.1980364968039670   \n69                 69  2.2057453301187055 2.2042644948848578   \n70                 70  2.2121039419929978 2.2106181772267641   \n71                 71  2.2185902235657848 2.2170994060694826   \n72                 72  2.2252060976812040 2.2237101011794951   \n73                 73  2.2319535468178571 2.2304522418363639   \n74                 74  2.2388346151750458 2.2373278689133587   \n75                 75  2.2458514108588186 2.2443390870575763   \n76                 76  2.2530061081734685 2.2514880669751598   \n77                 77  2.2603009500244813 2.2587770478276101   \n78                 78  2.2677382504393728 2.2662083397455794   \n79                 79  2.2753203972132576 2.2737843264669904   \n80                 80  2.2830498546865123 2.2815074681067840   \n81                 81  2.2909291666623672 2.2893803040661318   \n82                 82  2.2989609594728613 2.2974054560894777   \n83                 83  2.3071479452021597 2.3055856314783973   \n84                 84  2.3154929250769052 2.3139236264719010   \n85                 85  2.3239987930339860 2.3224223298035089   \n86                 86  2.3326685394768525 2.3310847264461940   \n87                 87  2.3415052552323639 2.3399139015571144   \n88                 88  2.3505121357210288 2.3489130446349558   \n89                 89  2.3596924853545147 2.3580854539036813   \n90                 90  2.3690497221753444 2.3674345409375510   \n\n         Gauss Method         Scipy Quad   Scipy Fixed Quad  \n0  2.0070899231544930 2.0070899231544930 2.0070899231544930  \n1  2.0071281359809907 2.0071281359809907 2.0071281359809738  \n2  2.0072427824638450 2.0072427824638450 2.0072427824635763  \n3  2.0074338866179300 2.0074338866179300 2.0074338866165680  \n4  2.0077014884840221 2.0077014884840221 2.0077014884797166  \n5  2.0080456441528005 2.0080456441528005 2.0080456441422858  \n6  2.0084664257984866 2.0084664257984866 2.0084664257766738  \n7  2.0089639217221720 2.0089639217221715 2.0089639216817443  \n8  2.0095382364048842 2.0095382364048842 2.0095382363358860  \n9  2.0101894905704585 2.0101894905704585 2.0101894904598931  \n10 2.0109178212583028 2.0109178212583023 2.0109178210897238  \n11 2.0117233819061293 2.0117233819061293 2.0117233816592472  \n12 2.0126063424427851 2.0126063424427851 2.0126063420930760  \n13 2.0135668893912779 2.0135668893912779 2.0135668889096094  \n14 2.0146052259821419 2.0146052259821423 2.0146052253344178  \n15 2.0157215722772830 2.0157215722772830 2.0157215714241148  \n16 2.0169161653044680 2.0169161653044685 2.0169161642008779  \n17 2.0181892592026327 2.0181892592026331 2.0181892577978062  \n18 2.0195411253781907 2.0195411253781907 2.0195411236152814  \n19 2.0209720526725592 2.0209720526725592 2.0209720504885658  \n20 2.0224823475411142 2.0224823475411151 2.0224823448668396  \n21 2.0240723342438125 2.0240723342438121 2.0240723310039246  \n22 2.0257423550477349 2.0257423550477354 2.0257423511609551  \n23 2.0274927704418286 2.0274927704418282 2.0274927658212598  \n24 2.0293239593641226 2.0293239593641221 2.0293239539177481  \n25 2.0312363194417378 2.0312363194417378 2.0312363130731184  \n26 2.0332302672440128 2.0332302672440128 2.0332302598532150  \n27 2.0353062385490936 2.0353062385490936 2.0353062300338780  \n28 2.0374646886243548 2.0374646886243548 2.0374646788816682  \n29 2.0397060925210457 2.0397060925210466 2.0397060814488568  \n30 2.0420309453835723 2.0420309453835723 2.0420309328830952  \n31 2.0444397627738526 2.0444397627738522 2.0444397487522110  \n32 2.0469330810112090 2.0469330810112090 2.0469330653845921  \n33 2.0495114575282884 2.0495114575282889 2.0495114402256585  \n34 2.0521754712435230 2.0521754712435230 2.0521754522109354  \n35 2.0549257229506743 2.0549257229506743 2.0549257021562881  \n36 2.0577628357260482 2.0577628357260487 2.0577628131658816  \n37 2.0606874553539765 2.0606874553539765 2.0606874310585028  \n38 2.0637002507712050 2.0637002507712054 2.0637002248128637  \n39 2.0668019145308785 2.0668019145308789 2.0668018870325908  \n40 2.0699931632868203 2.0699931632868203 2.0699931344315976  \n41 2.0732747382988665 2.0732747382988665 2.0732747083406182  \n42 2.0766474059600455 2.0766474059600455 2.0766473752356815  \n43 2.0801119583464427 2.0801119583464431 2.0801119272893764  \n44 2.0836692137906305 2.0836692137906310 2.0836691829457985  \n45 2.0873200174795938 2.0873200174795938 2.0873199875200985  \n46 2.0910652420781366 2.0910652420781375 2.0910652138236325  \n47 2.0949057883788029 2.0949057883788034 2.0949057628157415  \n48 2.0988425859794066 2.0988425859794071 2.0988425642832595  \n49 2.1028765939893264 2.1028765939893268 2.1028765775489009  \n50 2.1070088017657778 2.1070088017657786 2.1070087922097418  \n51 2.1112402296813602 2.1112402296813610 2.1112402289070884  \n52 2.1155719299242213 2.1155719299242217 2.1155719401290618  \n53 2.1200049873322935 2.1200049873322939 2.1200050110473505  \n54 2.1245405202630971 2.1245405202630976 2.1245405603896175  \n55 2.1291796815007356 2.1291796815007364 2.1291797413491564  \n56 2.1339236592017583 2.1339236592017587 2.1339237425334714  \n57 2.1387736778816953 2.1387736778816966 2.1387737889535390  \n58 2.1437309994441565 2.1437309994441578 2.1437311430556405  \n59 2.1487969242545004 2.1487969242545013 2.1487971057977071  \n60 2.1539727922602014 2.1539727922602028 2.1539730177722811  \n61 2.1592599841601632 2.1592599841601641 2.1592602603782876  \n62 2.1646599226253653 2.1646599226253658 2.1646602570439284  \n63 2.1701740735733699 2.1701740735733703 2.1701744745031806  \n64 2.1758039474993680 2.1758039474993693 2.1758044241284704  \n65 2.1815511008666122 2.1815511008666135 2.1815516633222947  \n66 2.1874171375592555 2.1874171375592568 2.1874177969706841  \n67 2.1934037104007955 2.1934037104007973 2.1934044789615941  \n68 2.1995125227415402 2.1995125227415415 2.1995134137714873  \n69 2.2057453301187038 2.2057453301187060 2.2057463581235561  \n70 2.2121039419929960 2.2121039419929973 2.2121051227212503  \n71 2.2185902235657835 2.2185902235657853 2.2185915740610018  \n72 2.2252060976812023 2.2252060976812040 2.2252076363282516  \n73 2.2319535468178548 2.2319535468178575 2.2319552933811648  \n74 2.2388346151750436 2.2388346151750462 2.2388365908266739  \n75 2.2458514108588163 2.2458514108588186 2.2458536381937866  \n76 2.2530061081734658 2.2530061081734685 2.2530086112094034  \n77 2.2603009500244791 2.2603009500244817 2.2603037541822300  \n78 2.2677382504393702 2.2677382504393728 2.2677413825007329  \n79 2.2753203972132554 2.2753203972132581 2.2753238852514581  \n80 2.2830498546865092 2.2830498546865123 2.2830537279644831  \n81 2.2909291666623641 2.2909291666623672 2.2909334554931822  \n82 2.2989609594728582 2.2989609594728613 2.2989656950360238  \n83 2.3071479452021562 2.3071479452021602 2.3071531593086028  \n84 2.3154929250769012 2.3154929250769056 2.3154986498747094  \n85 2.3239987930339816 2.3239987930339860 2.3240050606458675  \n86 2.3326685394768485 2.3326685394768529 2.3326753815594170  \n87 2.3415052552323599 2.3415052552323639 2.3415127024460078  \n88 2.3505121357210239 2.3505121357210288 2.3505202170981216  \n89 2.3596924853545094 2.3596924853545151 2.3597012275521689  \n90 2.3690497221753386 2.3690497221753448 2.3690591485976480  "
                    },
                    "metadata": {},
                    "output_type": "display_data"
                }
            ],
            "source": [
                "m = 1\n",
                "l = 1\n",
                "g = 9.8\n",
                "N = 1000\n",
                "theta_0_degrees_ls = np.linspace(0,90, 91)\n",
                "theta_0_ls = theta_0_degrees_ls * np.pi / 180 \n",
                "\n",
                "integral_func = lambda phi: (1-(np.sin(theta_0/2))**2*(np.sin(phi))**2)**(-.5)\n",
                "\n",
                "T= lambda int_func_eval: 4 * (l/g)**(.5) * int_func_eval\n",
                "\n",
                "data_to_display = []\n",
                "\n",
                "for i in range(len(theta_0_ls)):\n",
                "    theta_0 = theta_0_ls[i]\n",
                "    gauss_eval = T(gauss(integral_func,0, np.pi/2,N)) # integral_func should still grab the correct theta_0\n",
                "    qtrap_eval = T(qtrap(integral_func,0, np.pi/2,N))\n",
                "    qsimp_eval = T(qsimp(integral_func,0, np.pi/2,N+1)) # we use N+1 to make it odd-\u003erequired by Simpson's Law\n",
                "    scipy_quad = T(integrate.quad(integral_func, 0, np.pi/2)[0])\n",
                "    scipy_fixed_quad = T(integrate.fixed_quad(integral_func, 0, np.pi/2)[0])\n",
                "    data_to_display.append([int(theta_0_degrees_ls[i]), qtrap_eval, qsimp_eval, gauss_eval, scipy_quad, scipy_fixed_quad])\n",
                "\n",
                "df = pd.DataFrame(data_to_display, columns = ['theta_0 (degrees)', 'Trapezoidal Method', \"Simpson's Rule\", \"Gauss Method\", \"Scipy Quad\", \"Scipy Fixed Quad\"])\n",
                "\n",
                "pd.options.display.float_format = \"{:,.16f}\".format\n",
                "pd.set_option('display.max_rows', None)\n",
                "print(\"Comparing the Results of Different Integration Methods using N=1000 for Ours:\")\n",
                "display(df)"
            ]
        },
        {
            "cell_type": "code",
            "execution_count": 0,
            "metadata": {},
            "outputs": [],
            "source": [
                ""
            ]
        }
    ]
}