{
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   "source": [
    "%matplotlib inline\n",
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "import pandas as pd\n",
    "import sympy as sy"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "# Warm Up\n",
    "\n",
    "Below is a row from our heads tossing table.  Additionally we have built a plot of each outcome.  Please use the graph to answer the following questions.\n",
    "\n",
    "<center><img src = images/wrmup_d6.png /></center>\n",
    "\n",
    "1. How many ways are there to toss 0 heads?\n",
    "2. How many ways are there to toss 1 head?\n",
    "3. How many ways are there to toss 2 heads?\n",
    "4. How many total outcomes are possible?\n",
    "5. What is the probability of tossing 0 heads?\n",
    "6. What is the probability of tossing 1 head?\n",
    "7. What is the probability of tossing 2 heads?\n",
    "8. What is the probability of tossing 0 or 1 head?\n",
    "9. What is the probability of tossing 0, 1, or 2 heads?\n",
    "10. What does this have to do with area?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Riemann Review\n",
    "\n",
    "1. Below is a graph of the function $f(x) = -x^2 + 1$.  You are asked to approximate the area under the curve using the four rectangles shown.\n",
    "\n",
    "<center><img src = images/p1d6.png /></center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "slide"
    }
   },
   "source": [
    "2. Now we have the function $f(x) = x^2 - 2$.  Same problem using 8 rectangles.\n",
    "\n",
    "<center><img src = images/p2d6.png /></center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Getting Formal $\\Sigma$\n",
    "\n",
    "Understanding Summation Notation.  \n",
    "\n",
    "1. $\\sum_{i = 1}^n i = 1 + 2 + 3 + 4 + \\dots + (n - 1) + n$\n",
    "\n",
    "2. $\\sum_{i = 1}^n i^3 = 1^3 + 2^3 + 3^3 + 4^3 + \\dots + (n - 1)^3 + n^3$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
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     "slide_type": "slide"
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   },
   "source": [
    "**PROBLEM**\n",
    "\n",
    "Recall from last class, we had the fact that:\n",
    "\n",
    "1. $$\\sum_{i = 1}^n i = 1/2(n^2 + n)$$\n",
    "\n",
    "2. $$\\sum_{i = 1}^n i^3 = (1/2(n^2 + n))^2$$\n",
    "\n",
    "What is $\\sum_{i = 1}^5 i^3 - i$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### An Algorithm\n",
    "\n",
    "A Riemann sum $S$ of $f$ over $I$ with partition $P$ is defined as\n",
    "\n",
    "$$\\displaystyle S=\\sum _{i=1}^{n}f(x_{i}^{*})\\,\\Delta x_{i}$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "fragment"
    }
   },
   "source": [
    "Set up Riemann sum for $f(x) = x^3$ on $x \\in [0, 2]$. \n",
    "\n",
    "- What is $\\Delta x_i$?\n",
    "- What is $f(x_i^*)$?"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### A Definition\n",
    "\n",
    "\n",
    "\n",
    "The Definite Integral of $f(x)$ on $[a, b]$ is defined as:\n",
    "\n",
    "$$\\int_a^b f(x) dx = \\lim_{n \\to \\infty} \\sum_{i = 1}^n f(x_i^*)\\Delta x_i$$\n",
    "\n",
    "<center>\n",
    "<img src = images/rda2g1.gif />\n",
    "    </center>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### Using the Definition\n",
    "\n",
    "There are important patterns that emerge as a result of summations that we are able to capitalize on.  We've seen two, and will see how we extend these to general polynomials below.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
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   },
   "outputs": [],
   "source": [
    "i, n = sy.symbols('i n')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      " 2    \n",
      "n    n\n",
      "── + ─\n",
      "2    2\n",
      "=======================================\n",
      " 3    2    \n",
      "n    n    n\n",
      "── + ── + ─\n",
      "3    2    6\n",
      "=======================================\n",
      " 4    3    2\n",
      "n    n    n \n",
      "── + ── + ──\n",
      "4    2    4 \n",
      "=======================================\n",
      " 5    4    3     \n",
      "n    n    n    n \n",
      "── + ── + ── - ──\n",
      "5    2    3    30\n",
      "=======================================\n",
      " 6    5      4    2\n",
      "n    n    5⋅n    n \n",
      "── + ── + ──── - ──\n",
      "6    2     12    12\n",
      "=======================================\n"
     ]
    }
   ],
   "source": [
    "for exponent in np.arange(1, 6):\n",
    "    sy.pprint(sy.summation(i**exponent, (i, 1, n)))\n",
    "    print('=======================================')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "slide"
    }
   },
   "source": [
    "### A General Rule\n",
    "\n",
    "$$\\int_a^b x^n = \\frac{x^{n+1}}{n+1} |_{a}^b$$"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "slideshow": {
     "slide_type": "subslide"
    }
   },
   "source": [
    "### Some Examples\n",
    "\n",
    "1. $\\displaystyle \\int_0^3 x^2 dx$\n",
    "\n",
    "2. $\\displaystyle \\int_1^5 x^3 - x dx $\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### Net Change\n",
    "\n",
    "Suppose the velocity of a particle in motion is given by $v(t) = 3t - 5$ for $t \\in [0, 3]$.  Draw a Riemann sum approximation with 3 rectangles.  What would these areas mean?\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "import name_caller"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 17,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'What do you think Katrina'"
      ]
     },
     "execution_count": 17,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "name_caller.your_turn()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "# import gif\n",
    "\n",
    "# def f(x): return -x**2 + 2\n",
    "# x = np.linspace(-2, 2, 1000)\n",
    "\n",
    "# @gif.frame\n",
    "# def plot_riemann(n):\n",
    "#     a = x[0]\n",
    "#     b = x[-1]\n",
    "#     width = (b-a)/n\n",
    "#     plt.plot(x, f(x), color = 'black')\n",
    "#     bases = np.array([a + width*i for i in range(n)])\n",
    "#     plt.bar(bases, f(bases), width = width, align = 'edge', \n",
    "#            color = 'orange', edgecolor = 'black')\n",
    "#     areas = [width * height for height in f(bases)]\n",
    "#     plt.title(f'Area: {sum(areas)}')\n",
    "    \n",
    "# frames = []\n",
    "# for i in range(1, 1000,10):\n",
    "#     frame = plot_riemann(i)\n",
    "#     frames.append(frame)\n",
    "    \n",
    "# gif.save(frames, 'images/rda2g1.gif', duration = 200)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "###code for figure\n",
    "# N = 6\n",
    "# ks = np.arange(N+1)\n",
    "# outcomes = []\n",
    "# heads = []\n",
    "# for k in ks:\n",
    "#     outcomes.append(scipy.special.comb(N, k))\n",
    "#     heads.append(f'{k} heads')\n",
    "    \n",
    "# plt.figure(figsize = (15, 5))\n",
    "# plt.bar(ks, outcomes, color = '#EE340D', hatch = 'x', edgecolor = 'black', width = 1, alpha = 0.4)\n",
    "# plt.yticks(outcomes);\n",
    "# for count in outcomes:\n",
    "#     plt.axhline(count, color = 'black', linestyle = '--')\n",
    "# plt.xticks(ks, heads, rotation = 90);\n",
    "# plt.title(f'Number of ways for getting $x$ heads tossing {N} coins', fontsize = 16, loc = 'left');\n",
    "# plt.tight_layout()\n",
    "# plt.savefig('images/wrmup_d6.png')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "# def f(x): return -x**2 + 1\n",
    "# x = np.linspace(-1.1, 1.1, 1000)\n",
    "\n",
    "# fig, ax = plt.subplots(figsize = (12, 8))\n",
    "# ax.plot(x, f(x), color = 'black', alpha = 0.6)\n",
    "# ax.set_aspect('equal')\n",
    "# ax.grid(True, which='both')\n",
    "\n",
    "# # set the x-spine (see below for more info on `set_position`)\n",
    "# ax.spines['left'].set_position('zero')\n",
    "\n",
    "# # turn off the right spine/ticks\n",
    "# ax.spines['right'].set_color('none')\n",
    "# ax.yaxis.tick_left()\n",
    "\n",
    "# # set the y-spine\n",
    "# ax.spines['bottom'].set_position('zero')\n",
    "\n",
    "# # turn off the top spine/ticks\n",
    "# ax.spines['top'].set_color('none')\n",
    "# ax.xaxis.tick_bottom()\n",
    "# bars = [-1, -0.5, 0, 0.5]\n",
    "# for bar in bars:\n",
    "#     ax.bar(bar, f(bar), width = 0.5, align = 'edge', color = 'orange', edgecolor = 'black', alpha = 0.7)\n",
    "#     ax.text(bar + 0.05, f(bar)/2, f'Rectangle {bars.index(bar)}\\nHeight = {f(bar)}', fontsize = 14)\n",
    "# ax.set_title('Approximating area under $f(x) = -x^2 + 1$ with 4 rectangles.', fontsize = 14, loc = 'left')\n",
    "# plt.savefig('images/p1d6.png')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {
    "slideshow": {
     "slide_type": "skip"
    }
   },
   "outputs": [],
   "source": [
    "# def f(x): return x**2 - 2\n",
    "# x = np.linspace(0, 2, 1000)\n",
    "\n",
    "# fig, ax = plt.subplots(figsize = (20, 8))\n",
    "# ax.plot(x, f(x), color = 'black', alpha = 0.6)\n",
    "# #ax.set_aspect('equal')\n",
    "# ax.grid(True, which='both')\n",
    "\n",
    "# # set the x-spine (see below for more info on `set_position`)\n",
    "# ax.spines['left'].set_position('zero')\n",
    "\n",
    "# # turn off the right spine/ticks\n",
    "# ax.spines['right'].set_color('none')\n",
    "# ax.yaxis.tick_left()\n",
    "\n",
    "# # set the y-spine\n",
    "# ax.spines['bottom'].set_position('zero')\n",
    "\n",
    "# # turn off the top spine/ticks\n",
    "# ax.spines['top'].set_color('none')\n",
    "# ax.xaxis.tick_bottom()\n",
    "# width = 1/4\n",
    "# bars = [0 + i*width for i in range(8)]\n",
    "# for bar in bars:\n",
    "#     ax.bar(bar, f(bar), width = 1/4, align = 'edge', color = 'orange', edgecolor = 'black', alpha = 0.7)\n",
    "#     ax.text(bar, f(bar)/2, f'Rectangle {bars.index(bar) + 1}\\nHeight = {f(bar):.2f}', fontsize = 14)\n",
    "# ax.set_title(f'Approximating area under $f(x) = x^2 - 2$ with {len(bars)} rectangles.', fontsize = 20, loc = 'left')\n",
    "# plt.savefig('images/p2d6.png')"
   ]
  }
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