Fundamental Theorem of Calculus.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# MSDS 400 Fundamental Theorem of Calculus"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "# Sample function\n",
    "\n",
    "def f(x):\n",
    "    f = 1997 + 15*x**30  # Students can supply a function at this point.\n",
    "    return f"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "`integrate()` is a general numerical integration function. It requires an interval [a,b] and n = the number of subintervals used for integration. The code below uses the function, f, as defined f above.  "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "def integrate(a, b, n):\n",
    "    total = 0.0\n",
    "    delta = (b - a) / n\n",
    "    for i in range(n):\n",
    "        total = total + delta * (f(a + delta * (i + 1)) + f(a + delta * i)) / 2\n",
    "    return total\n",
    "\n",
    "# This defines the parameters for integration.\n",
    "\n",
    "c = 4.0\n",
    "b = 24.0\n",
    "a = 0.0\n",
    "n = 100"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "One area is negative and the other is positive.  We integrate them separately\n",
    "and combine their absolute values."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Final Estimate of Total Profits = 5.958637284756276e+42\n"
     ]
    }
   ],
   "source": [
    "area1 = integrate(a, b, n)\n",
    "area2 = integrate(b, c, n)\n",
    "area = abs(area1) + np.abs(area2)\n",
    "\n",
    "print(\"Final Estimate of Total Profits = {}\".format(area))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The next section of code shows how to plot different colors by dividing the interval based on which area is negative.\n",
    "\n",
    "`fill_between()` requires an array for x and two functions for y between which the color is filled. In this case we use 0.0 for the x-axis and y. `alpha` controls the intensity of the color.\n",
    "\n",
    "This section sets up the dimensions of the plot and creates it. We have to consider both plots and find the min and max of y for plot limits."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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AUyS1AZ8DxgNExJDnBczMyqpMndJAjokgIs5qoOw5ecVhZjbSynSiGHxnsZnZsOvuLk+zEDgRmJkNuzL1TgZOBGZmw667G/r6io6ifk4EZmbDzInAzKzF9fSUp3cycCIwMxt2nZ3l6a8YnAjMzIbdli3luo/AicDMbJht3er7CMzMWlqZOqUBJwIzs2G3bZubhszMWlpXl2sEZmYtzYnAzKzFlalTGnAiMDMbVn19sGOHawRmZi1rx4705FE/fdTMrEX19MC4IjsB3gu5JQJJiyRtknT/AOPfJ+m+7O8uScfnFYuZ2UgpW6c0kG+NYDEwZ5Dxa4A3RMRxwOeBhTnGYmY2IsrWKQ3k21XlUkmzBhl/V9Xg3cD0vGIxMxspZeuUBkbPOYLzgB8PNFLSPEnLJS3fvHnzCIZlZtaY7u5yPXkURkEikPRGUiL4zEBlImJhRMyOiNlTp04dueDMzBpUtk5pIMemoXpIOg64GpgbEU8UGYuZ2XDo7i5XpzRQYI1A0kzgBuADEfHbouIwMxtO27a5RvAMSdcCpwBTJLUBnwPGA0TEAuAy4HnAVUqn2HsjYnZe8ZiZjYQtW8p3+WieVw2dNcT484Hz85q/mVkRytYpDYyCk8VmZs2ks7NcD5wDJwIzs2G1bZtrBGZmLc2JwMysxXV1uWnIzKyl9fS4RmBm1rIinAjMzFrazp0pGTgRmJm1qO7u8nVKA04EZmbDpoyd0oATgZnZsHEiMDNrcWXslAacCMzMhk0ZO6UBJwIzs2HT01O+R1CDE4GZ2bDp7obdu4uOonFOBGZmw6Sry4nAzKylbd0KqZ+tcsktEUhaJGmTpPsHGC9JX5W0StJ9kv4gr1jMzEbC1q3le+Ac5FsjWAzMGWT8XOCY7G8e8PUcYzEzy10ZeyeDHBNBRCwFtgxS5AzgG5HcDUyWdHhe8ZiZ5e3JJ50IGjUNWF813Ja9twdJ8yQtl7R88+bNIxKcmVmjtm1z01Cj+jul0u+tGBGxMCJmR8TsqVOn5hyWmdne6epyjaBRbcCMquHpQHtBsZiZ7TM/a6hxNwJnZ1cPvRp4MiI6CozHzGyf9PSUs2kotydnS7oWOAWYIqkN+BwwHiAiFgBLgNOAVcB24Ny8YjEzy1tvL+zaVc77CHJLBBFx1hDjA/hoXvM3MxtJlU5pypgIhmwakvRlSYdIGi/p55Iel/T+kQjOzKwsynp+AOo7R/AnEbENeAvpBO+LgU/lGpWZWck0eyIYn/0/Dbg2Iga7SczMrCV1d5ezWQjqO0fwI0kPATuAj0iaCjyVb1hmZuXS01POTmmgjhpBRFwCvAaYHRG7gB7S4yHMzCxT1t7JYJAagaRTI+JWSX9a9V51kRvyDMzMrEzK2ikNDN409AbgVuCt/YwLnAjMzJ7RlIkgIj6X/feNXmZmQ+jsLDqCvVfPfQTflDSpavhIST/PNywzs3Ipa18EUN/lo3cC90g6TdIFwC3AFblGZWZWMmVOBENePhoR/yxpJXAb8Djw+xGxIffIzMxKpLOznA+cg/qahj4ALALOJnU/uUTS8TnHZWZWKtu2NXGNAHgn8LqI2ARcK+n7wL8DJ+QZmJlZmXR1lbdGUE/T0Ntrhn8t6aTcIjIzK6EyP2toyEQgaQJwHvByYELVqA/lFZSZWdl0d8PEiUVHsXfqyV/fBF4IvBm4g9SlZFeeQZmZlUlfHzz9dHlrBPWEfXREXAr0RMS/A6cDr6xn4pLmSHpY0ipJl/QzfpKkH0n6jaSVknzzmpmVTk9PSgJlffpoPYlgV/a/U9IrgEnArKE+JGkscCUwFzgWOEvSsTXFPgo8EBHHk7q1/AdJ+9UXupnZ6NDdXd4TxVBfIlgo6VDgs6QO5x8AvlTH504CVkXEIxGxE7iOPZ9aGsDBSk+zOwjYAvTWG7yZ2WjQ1VXeZiGo76qhq7OXS4Hfa2Da04D1VcNtwMk1Zb5GSi7twMHAeyKir3ZCkuYB8wBmzpzZQAhmZvnbuLHoCPZNQzlM0k2NFO/nvdqndb8ZuBc4gnRfwtckHbLHhyIWRsTsiJg9derUBkIwM8tfR0d5nzwKDSYC0lF+vdqAGVXD00lH/tXOBW6IZBWwBnhpgzGZmRVq/XrYtWvocqNVPY+YuEjS5GzwvxqY9jLgGElHZSeAzyQ1A1VbB7wpm88LgJcAjzQwDzOzwq1aBePHD11utKrnERMvBJZL+k9gkSRFDN0hW0T0SroIuBkYCyyKiJWSLszGLwA+DyyW9N+kpqTPRMTje7swZmZFePTRJk8EEfFZSZcCf0JqyvmapOuBf42I1UN8dgmwpOa9BVWv27PpmpmVVnt7uRNBXecIshrAhuyvFzgU+K6kL+cYm5nZqLdzZ3ry6Lh62ldGqXqeNXQx8EFSXwRXA5+KiF2SxgC/Az6db4hmZqPXhg2pNlDWu4qhvnMEU4A/jYhHq9+MiD5Jb8knLDOzcujoKPfNZFDfOYLLBhn34PCGY2ZWLu3t5b6HABq/j8DMzKq0taXzBGXmRGBmtg8eeaTcJ4rBicDMbJ+sXVvuS0fBicDMbJ+0tTkRmJm1rN274YknnAjMzFrWpk3p/ECZ7yEAJwIzs73W0VHunskqnAjMzPbShg2p4/qycyIwM9tLjz1W/nsIwInAzGyvrV5d/sdLgBOBmdleW7Om/FcMgROBmdlee+wxJ4IhSZoj6WFJqyRdMkCZUyTdK2mlpDvyjMfMbLj09cHGjc2RCHJ7QoakscCVwB+TOrJfJunGiHigqsxk4CpgTkSsk/T8vOIxMxtOW7em/z5HMLiTgFUR8UhE7ASuA86oKfNe4IaIWAcQEZtyjMfMbNh0dJT/YXMVeSaCacD6quG27L1qLwYOlXS7pBWSzu5vQpLmSVouafnmzZtzCtfMrH4dHRBRdBTDI89E0N9N17WrbRzwKuB04M3ApZJevMeHIhZGxOyImD116tThj9TMrEEdHbBrV9FRDI88KzZtwIyq4elAez9lHo+IHqBH0lLgeOC3OcZlZrbP1qxxjaAey4BjJB0laT/gTODGmjI/BF4vaZykA4CTAXd/aWaj3urVzXHFEORYI4iIXkkXATcDY4FFEbFS0oXZ+AUR8aCknwD3AX3A1RFxf14xmZkNl/XrnQjqEhFLgCU17y2oGb4cuDzPOMzMhlNEOkcwcWLRkQyPJrgC1sxsZHV1pRPFzXAPATgRmJk1bMOG1CxU9g5pKpwIzMwa1NHRPEkAnAjMzBrW0dEc/RBUOBGYmTVo7drm6JmswonAzKxBjzzSPJeOghOBmVnD1q1zIjAza2nt7U4EZmYta8eOdB/B2LFFRzJ8nAjMzBqwYQPst58vHzUza1kdHc1zR3FFky2OmVm+NmyA3t6ioxheTgRmZg1Yt655OqSpcCIwM2tAM/VDUOFEYGbWgLVrnQjMzFpaR4cTQUMkzZH0sKRVki4ZpNyJknZLelee8ZiZ7Ytdu2DLFhiXa5deIy+3RCBpLHAlMBc4FjhL0rEDlPsSqUtLM7NRa9Om5uqHoCLPGsFJwKqIeCQidgLXAWf0U+5jwPeATTnGYma2z5rxHgLINxFMA9ZXDbdl7z1D0jTgHcBz+jE2MxuNOjpg9+6ioxh+eSaC/ipPUTN8BfCZiBh01UqaJ2m5pOWbN28ervjMzBry2GPN1SFNRZ6nPNqAGVXD04H2mjKzgeuUGtymAKdJ6o2IH1QXioiFwEKA2bNn1yYTM7MRsXp1850ohnwTwTLgGElHAY8BZwLvrS4QEUdVXktaDNxUmwTMzEaLNWua79JRyDERRESvpItIVwONBRZFxEpJF2bjfV7AzEql2Tqkqci1khMRS4AlNe/1mwAi4pw8YzEz2xcdHekegkmTio5k+DXhhVBmZsNv2bJ06Wiz3UMATgRmZnW57bbmvHQUnAjMzIYUAXfcAQceWHQk+XAiMDMbwtq10N3dnCeKwYnAzGxIy5ZBX19znh8AJwIzsyH97GfNmwTAicDMbFB9fXDXXc17fgCcCMzMBvXQQ6kfgmZ8tESFE4GZ2SDuuQd6e4uOIl9OBGZmg7jlluauDYATgZnZgHbuhBUr4IADio4kX04EZmYDuP/+dLXQ2LFFR5IvJwIzswHcfXdzdkRTy4nAzGwAP/0p7L9/0VHkz4nAzKwf27enpqGJE4uOJH9OBGZm/bj33nRuYEwL7CVzXURJcyQ9LGmVpEv6Gf8+Sfdlf3dJOj7PeMzM6nXnnelGslaQWyKQNBa4EpgLHAucJenYmmJrgDdExHHA58k6qDczK9ott8CECUVHMTLyrBGcBKyKiEciYidwHXBGdYGIuCsitmaDdwPTc4zHzKwunZ2po/pWOD8A+SaCacD6quG27L2BnAf8OMd4zMzqsmJFOj/QzE8crZbnjdP9rcLot6D0RlIieN0A4+cB8wBmzpw5XPGZmfVr6dLWOT8A+dYI2oAZVcPTgfbaQpKOA64GzoiIJ/qbUEQsjIjZETF76tSpuQRrZlZx663N/1iJankmgmXAMZKOkrQfcCZwY3UBSTOBG4APRMRvc4zFzKwuGzdCR0dr3EhWkVvTUET0SroIuBkYCyyKiJWSLszGLwAuA54HXKXUGNcbEbPzisnMbCjLlqV7B1rl/ADke46AiFgCLKl5b0HV6/OB8/OMwcysXhFwzTXpfytpgXvmzMzqc8cdsHw5HHJI0ZGMLCcCMzPSU0YvvRTGj2+tZiFwIjAzA+Daa9NJ4mbupH4gTgRm1vK2boXLL2+dO4lrORGYWcv7ylfgqada65LRak4EZtbSVq1KzUIHH1x0JMVxIjCzlhUB8+en/83eL/FgnAjMrGX94hepX+JJk4qOpFhOBGbWknbtgs9+FsaNa73LRWs5EZhZS7ruOmhvh4MOKjqS4jkRmFnLefxx+PKXW/dy0VpOBGbWUv77v+G00+Dpp1v3ctFaTgRm1hIi4Prr4Z3vhCefbL3nCQ0m16ePmpmNBk8/DZddBt/9bnqExH77FR3R6OJEYGZNrb0dLrgAHnooXSY6xu0ge/AqMbOmFAG//CXMnQsPPwyTJzsJDMQ1AjNrKuvWwZIl6fLQ9evTCeHJk4uOanTLNRFImgP8I6mryqsj4u9qxisbfxqwHTgnIv4zz5jMrPl0dMDNN6dnBq1enWoDlQTQ6jeL1SO3RCBpLHAl8MdAG7BM0o0R8UBVsbnAMdnfycDXs/9mZs8RAZ2d6Yj/0UdhzRpYuRJ+97t05A+pU5lJk7zzb1SeNYKTgFUR8QiApOuAM4DqRHAG8I2ICOBuSZMlHR4RHTnGZZaLnTvhZz9LjzNuJpX+e6v78a19L+K5f5X3+vrS3+7dabjyv7c3ra9du577eudO2LYt9Q/Q2Zled3XtWW7s2DTd8ePTFUATJjwb22hb/729RUcwtDwTwTRgfdVwG3se7fdXZhrwnEQgaR4wLxvslvTw8IbauMPhlYdBCTZx/rbAOK8L2MHEMeuYtH8vz9tRdCyjw5ZxcFjle9HPMbr2eDFgUYKURvp2D1t4I+jii9c9dfHFO9YPXTJXRw40Is9EMNDWbLQMEbEQWDgcQQ0XScvbI2YXHcdo4HXxLEnLIzq8Lqisi3avCyrrYvT+RvK8mKoNmFE1PB1o34syZmaWozwTwTLgGElHSdoPOBO4sabMjcDZSl4NPOnzA2ZmIyu3pqGI6JV0EXAz6fLRRRGxUtKF2fgFwBLSpaOrSJePnptXPDkYVU1VBfO6eJbXxbO8Lp41qteFIvZokjczsxbiG67NzFqcE4GZWYtzImiQpDmSHpa0StIlRcdTJEmLJG2SdH/RsRRJ0gxJt0l6UNJKSR8vOqaiSJog6deSfpOti/9bdExFkzRW0n9JuqnoWAbiRNCAqsdmzAWOBc6SdGyxURVqMTCn6CBGgV7gExHxMuDVwEdb+HvxNHBqRBwPnADMya4IbGUfBx4sOojBOBE05pnHZkTETqDy2IyWFBFLgS1Fx1G0iOioPCwxIrpIP/ppxUZVjEi6s8Hx2V/LXpEiaTpwOnB10bEMxomgMQM9EsMMAEmzgN8H7ik4lMJkTSH3ApuAWyKiZdcFcAXwaaCv4DgG5UTQmLoeiWGtSdJBwPeA/x0R24qOpygRsTsiTiA9KeAkSa8oOKRCSHoLsCkiVhQdy1CcCBrjR2JYvySNJyWBb0XEDUXHMxpERCdwO617Hum1wNskrSU1I58q6ZpiQ+qfE0Fj6nlshrWYrIOlfwUejIj/V3Q8RZI0VdLk7PVE4I+AhwoNqiAR8ZcRMT0iZpH2FbdGxPsLDqtfTgQNiIheoPLYjAeB6yNiZbFRFUfStcCvgJdIapN0XtExFeS1wAdIR3z3Zn+nFR1UQQ4HbpN0H+nA6ZaIGLWXTVriR0yYmbU41wjMzFqcE4GZWYtzIjAza3FOBGZmLc6JwMysxTkRmI0ASRdKOrvoOMz648tHzcxanGsEZjUknSjpvuzZ+gdmz9V/RU2Zt0q6J3vO/M8kvSB7/6uSLstev1nSUkljJM2X9Mns/YslPZDN47qRX0Kz53KNwKwfkv4GmABMBNoi4os14w8FOiMiJJ0PvCwiPiHpANIdtRcBC4DTImK1pPlAd0T8vaR24KiIeFrS5OyZPGaFGVd0AGaj1F+TduhPARf3M3468B1JhwP7AWsAImK7pAuApcCfR8Tqfj57H/AtST8AfjD8oZs1xk1DZv07DDgIOBiYIOkLlecIZeP/CfhaRLwS+DCp9lDxSuAJ4IgBpn06qae7VwErJPmAzArlRGDWv4XApcC3gC9FxF9FxAnZc/YBJgGPZa8/WPmQpCOBT5A6p5kr6eTqiUoaA8yIiNtIHZZMJiUcs8L4SMSsRnaZZ29EfDvrp/ouSadGxK1VxeYD/yHpMeBu4Kiqx1F/MiLas6exLpZ0YtXnxgLXSJpE6ujoKz5HYEXzyWIzsxbnpiEzsxbnRGBm1uKcCMzMWpwTgZlZi3MiMDNrcU4EZmYtzonAzKzF/Q9buZLbaBNeXQAAAABJRU5ErkJggg==\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.arange(0.0, 2.1, 0.1)  # This defines the interval for the color red.\n",
    "y = f(x)\n",
    "\n",
    "x1 = np.arange(2.0, 4.1, 0.1)  # This defines the interval for the color blue.\n",
    "y1 = f(x1)\n",
    "\n",
    "plt.figure()\n",
    "plt.fill_between(x, 0, y, color='r', alpha=0.8)\n",
    "plt.fill_between(x1, 0.0, y1, color='b', alpha=0.8)\n",
    "\n",
    "ymin = min(min(y), min(y1))\n",
    "ymax = max(max(y), max(y1))\n",
    "plt.xlim(-0.5, 4.5)\n",
    "plt.ylim(ymin - 1.0, ymax + 1.0)\n",
    "\n",
    "plt.xlabel('x-axis')\n",
    "plt.ylabel('y-axis')\n",
    "plt.title('Plot Showing Color Coded Integration Areas')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "This section of code shows how to plot different colors using a\n",
    "logical operator in the plotting statement.  The operator will\n",
    "control when each color is used."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "plt.figure()  # This separates the two plots from each other.\n",
    "x = np.arange(0.0, 4.1, 0.1)  # This defines the interval for color filling.\n",
    "y = f(x)\n",
    "plt.plot(x, y, c='k')\n",
    "\n",
    "# This shows how to fill between the lines using an inequality.\n",
    "\n",
    "plt.fill_between(x, 0.0, y, where=y < 0.0, facecolor='y', interpolate=True)\n",
    "plt.fill_between(x, 0.0, y, where=y > 0.0, facecolor='g', interpolate=True)\n",
    "\n",
    "plt.xlim(-0.5, 4.5)\n",
    "plt.ylim(ymin - 1.0, ymax + 1.0)\n",
    "plt.xlabel('x-axis')\n",
    "plt.ylabel('y-axis')\n",
    "plt.title('Plot Showing Color Coded Integration Areas')\n",
    "plt.show()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
 ],
 "metadata": {
  "kernelspec": {
   "display_name": "Python 3",
   "language": "python",
   "name": "python3"
  },
  "language_info": {
   "codemirror_mode": {
    "name": "ipython",
    "version": 3
   },
   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
   "nbconvert_exporter": "python",
   "pygments_lexer": "ipython3",
   "version": "3.8.10"
  },
  "varInspector": {
   "cols": {
    "lenName": 16,
    "lenType": 16,
    "lenVar": 40
   },
   "kernels_config": {
    "python": {
     "delete_cmd_postfix": "",
     "delete_cmd_prefix": "del ",
     "library": "var_list.py",
     "varRefreshCmd": "print(var_dic_list())"
    },
    "r": {
     "delete_cmd_postfix": ") ",
     "delete_cmd_prefix": "rm(",
     "library": "var_list.r",
     "varRefreshCmd": "cat(var_dic_list()) "
    }
   },
   "types_to_exclude": [
    "module",
    "function",
    "builtin_function_or_method",
    "instance",
    "_Feature"
   ],
   "window_display": false
  }
 },
 "nbformat": 4,
 "nbformat_minor": 2
}