{
 "cells": [
  {
   "cell_type": "code",
   "execution_count": 2,
   "id": "3d760f8c-4538-4ba2-a543-2c04eec818d1",
   "metadata": {},
   "outputs": [],
   "source": [
    "import matplotlib.pyplot as plt #as after a package indicates an alias you can use to call i\n",
    "import numpy as np\n",
    "from random import seed\n",
    "from random import random\n",
    "from random import uniform\n",
    "seed(6) # Random number seed"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "2f46cff3-0233-4e01-962b-b7a18d10dd43",
   "metadata": {},
   "source": [
    "# PROBLEM #1: \n",
    "## Integrate Sphere"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "35067bb7-1028-4ab1-8054-fc5378bf6bec",
   "metadata": {},
   "source": [
    "## Integrate Sphere: 2-D"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "id": "031bdb0b-4971-4e5d-8d6b-5451ea51b44d",
   "metadata": {},
   "outputs": [],
   "source": [
    "#set the number of points\n",
    "n = 10"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "beef83f7-19c8-48a2-8288-2c991bddaa89",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[-0.5047762 ,  0.9607155 , -0.30034046, -0.8839553 , -0.99124459,\n",
       "        -0.06122525, -0.99256951,  0.90379612, -0.87078747, -0.06361406]])"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "xpoints = 2*np.random.rand(1,n)-1\n",
    "ypoints = 2*np.random.rand(1,n)-1\n",
    "xpoints"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "id": "3d9570e7-fae0-429a-b4d0-c0b92e4b8df0",
   "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": [
    "theta = np.linspace(0,2*3.14, 100)\n",
    "z = np.linspace(0, 2*3.14, 100)\n",
    "r = 1\n",
    "x = r * np.sin(theta)\n",
    "y = r * np.cos(theta)\n",
    "\n",
    "\n",
    "plt.axes().set_aspect('equal')\n",
    "plt.plot(x,y,color='red');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "63ada214-b33c-4412-9917-33193ef4aad1",
   "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.axes().set_aspect('equal')\n",
    "plt.plot(x,y,color=\"red\")\n",
    "plt.plot(xpoints,ypoints,'o',color='black');\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "id": "15d033bd-f0f6-41fa-992a-5cea1b71af53",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.46683698, 0.92682248, 0.27007036, 0.78344335, 0.98748397,\n",
       "        0.58469285, 1.09971856, 0.83148623, 0.75868008, 0.80486978]])"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "distance=xpoints**2+ypoints**2\n",
    "distance"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "id": "e6953317-4a0b-4c7a-abca-aea652890eed",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ True,  True,  True,  True,  True,  True, False,  True,  True,\n",
       "         True]])"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "distance<=1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "id": "67b9e19a-ce39-4801-8ed8-6f37f42f623c",
   "metadata": {
    "tags": []
   },
   "outputs": [],
   "source": [
    "count=np.count_nonzero(distance<=1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "id": "602492c2-3e71-475d-aed5-e49d34f33265",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "3.6 3.141592653589793 1.1459155902616465\n"
     ]
    }
   ],
   "source": [
    "result=count/n*4\n",
    "answer=np.pi\n",
    "print(result,answer,result/answer)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "id": "10683956-f39d-4454-9f73-4ec6805fec50",
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "id": "4e1182d6-e6bf-4bba-9011-a67fed97fb58",
   "metadata": {},
   "source": [
    "# PROBLEM #2"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cdd627ae-feb4-4657-ba30-232f78510af8",
   "metadata": {},
   "source": [
    "## Integrate Sphere: 3-D"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "186f0b51-4835-49b7-b7b9-f1490b49fd54",
   "metadata": {},
   "source": [
    "## Integrate Sphere: 4-D"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "571df5cf-2c9c-40ea-961c-81a98061979f",
   "metadata": {},
   "source": [
    "## Integrate Sphere: 5-D"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "98cc9ec6-750e-4003-b46a-c5059440f5f4",
   "metadata": {},
   "source": [
    "# PROBLEM #3"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ee2b4ea0-5a46-4e97-a0d3-51f0c72fee3e",
   "metadata": {
    "tags": []
   },
   "source": [
    "# Integrate  Exp[-x**2]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "id": "98c36854-2680-49fb-a43f-1857b29e4f8b",
   "metadata": {},
   "outputs": [],
   "source": [
    "n=10\n",
    "seed(6)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "id": "b2615184-d0f9-46d4-8357-07353a1c9bbb",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.22027026705244174"
      ]
     },
     "execution_count": 12,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def func0(x):  # This is the real function\n",
    "     return np.exp(-x**2) \n",
    "    \n",
    "# In Homework problem #4, we'll tweak this \n",
    "# In Homework problem #3, func0() and func() are idential\n",
    "# \n",
    "def func(x):  \n",
    "     return func0(x)\n",
    "    \n",
    "func(1.23)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "id": "4bec823b-70aa-4c03-8091-5e5fd4bdb43d",
   "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": [
    "x = np.linspace(-0, 10, 100)\n",
    "y0 = func0(x)\n",
    "y =func(x)\n",
    "plt.plot(x,y0,color='red');\n",
    "plt.plot(x,y,color='blue');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "id": "99e97921-67a9-4e17-aa99-7b420ee0098f",
   "metadata": {},
   "outputs": [],
   "source": [
    "xpoints1 = np.random.rand(1,n)\n",
    "ypoints =  np.random.rand(1,n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "id": "45458d85-d169-4c8d-9475-c81a588d3794",
   "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.plot(xpoints1,ypoints,'o',color='black');\n",
    "plt.plot(x,y,color='red');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 18,
   "id": "130d2d11-e3df-4116-8a52-6cf073ac68cd",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[0.48140824, 0.399868  , 0.99614284, 0.75606364, 0.92302499,\n",
       "        0.53472557, 0.85369191, 0.94504301, 0.99844334, 0.99070363]])"
      ]
     },
     "execution_count": 18,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ychecks1 = func(xpoints1)\n",
    "ychecks1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 19,
   "id": "1eda68f8-35b0-4a42-bad3-f06cc83ad858",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "array([[ True,  True,  True,  True,  True, False, False,  True,  True,\n",
       "         True]])"
      ]
     },
     "execution_count": 19,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ypoints<=ychecks1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 20,
   "id": "283eebe4-7835-40f4-89c9-05ab46d7431c",
   "metadata": {
    "tags": []
   },
   "outputs": [
    {
     "data": {
      "text/plain": [
       "8"
      ]
     },
     "execution_count": 20,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "np.count_nonzero(ypoints<=ychecks1)"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "7fc8e632-1dfe-4fbc-93e9-84dbbf332cc6",
   "metadata": {},
   "source": [
    "# PROBLEM #4"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cd0e48fe-3f71-4212-98f0-eab9135ba77e",
   "metadata": {},
   "source": [
    "# Integrate #2"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "id": "71fa22b2-b116-4943-92e4-292c2bf38926",
   "metadata": {},
   "outputs": [],
   "source": [
    "n=100\n",
    "seed(1)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "id": "c888f253-4366-40d8-a961-f91d1fa96baf",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.8871446422000988"
      ]
     },
     "execution_count": 24,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "def func0(x):  # This is the real function\n",
    "     return np.exp(-(x/5)**2) * np.sin(x)\n",
    "    \n",
    "# In Homework problem #4, we'll tweak this \n",
    "# In Homework problem #3, func0() and func() are idential\n",
    "# \n",
    "def func(x):  \n",
    "     return func0(x) \n",
    "    \n",
    "func(1.23)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "id": "dfbed174-33df-4879-a0f8-2a84f0863c7b",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": 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ALJiWYTmJUsod3L7DWkQCgFeAJy63rDHmHWNMojEmMSYmxt3R/m3F7hh6191FcFiwx8b0RgmjWtE08BDzf/Dvr4NSvsoVhX8IaFrheWz5az+pA3QEkkVkH9AHmOMtB24LcgrYfLYN/TvowUoJEJLi9/DDkQSKzxTbjqOUcjFXFP56IF5EWohIMHArMOenN40xecaYaGNMc2NMc2ANMNoYk+KCsZ2W8mk6pQTRd2io7SheYcToYPIJZ9V7enqmUr7G6cI3xpQAU4CFwA5gljEmTUSeF5HRzq7f3VbNOwVAn4n+ef79hYZMTSCI8yz87JTtKEopFxNjjO0MlUpMTDQpKe7/JWB0w7XsPhnDznMt3T5WdTGo7mZOF4ew6Ww721GUUldIRDYYYyrdZe6fVxmVMw7DqpzW9Is7dPmF/UhS71NsLmxH9tYc21GUUi7k14Wf/v0+jpso+vW1ncS7JP2iIQCL3txtOYlSypX8uvBXf1G2Zd93nP9NmHYpXX7ehvqSy4JF/jVNtFK+zq8Lf9UKBxHk0f4G3X9fUUBQAMNb7GLRgXaUFvvfdNFK+Sq/LvzV+xrSJ8p/J0y7lKQk4biJYuOnu2xHUUq5iN82Xd6BPFKLWtOv8xnbUbzSsCltERws+FAP3CrlK/y28Nd+nIEhgH5J4bajeKWY9tH0qL2ThesjbUdRSrmI3xb+6kX5CA563dbadhSvldQ9hzX5CZzar9NOKOUL/LbwV20Lo1NIOuGxuoV/McMnRFJKED+8rtMsKOUL/LLwHSUO1h6Pp0/zo7ajeLU+dyUQQR4LvztvO4pSygX8svB3LdhLHhH06avnmV9KUEgQQxrvYEFGa4zDO6fgUEpVnV8W/pqvyu7o1GdcY8tJvF/SkGKyShuz49s9tqMopZzkl4W/do0hgjzaJrWwHcXrDX+gbBbRBe9nWU6ilHKWfxb+vvr0iszQC66qoFnfJrQP3sPClWG2oyilnOR3jVeQU8DWwnh6J+TbjlJtDO+QxdLjHTl77KztKEopJ/hd4W/4LAMHgfQZone4qqqk8aGcI4Rl/9DTM5Wqzvyu8NfMPwlAr1t0/31VDXygAyEUsuDLAttRlFJO8LvCX7slhFZB+4lpH207SrVRK7IWg6LSWLg91nYUpZQT/K7w12TH0buJnnFypZIGFLCzuBX7VujXTqnqyq8KP2v9EQ47GtEnscR2lGpn+N1lW/cL3860nEQpdbVcUvgikiQiu0QkQ0SequT9x0Vku4hsFZHFIhLninGv1JqZ+wDofYPuzrlS7Ua2pFlgFguWBNuOopS6Sk4XvogEAtOAEUACMEFEEi5YbBOQaIzpDHwB/NXZca/G2mXnCOYcXcbrDJlXSgKEpPhMFh9JoPhMse04Sqmr4Iot/F5AhjEm0xhTDMwExlRcwBizxBjz00ncawArR//W7K5H97Dd1AyvaWP4am/E2JrkE87q9/X0TKWqI1cUfhPgYIXnWeWvXczdwPzK3hCRySKSIiIpubm5Loj2f86fPc+G0/H0bn3Cpev1J9c91J4gzjP/01O2oyilroJHD9qKyO1AIvBSZe8bY94xxiQaYxJjYmJcOnba3EwKqU3va2q4dL3+JDw2nGvqprJgSyPbUZRSV8EVhX8IaFrheWz5a/9BRIYCzwCjjTHnXDDuFVk3t2zu+143XuqXD3U5SX3y2FLUlsMbs21HUUpdIVcU/nogXkRaiEgwcCswp+ICItIN+CdlZW/lrtjr1guRcoKWg5vZGN5nJP2ybOt+4ZvplpMopa6U04VvjCkBpgALgR3ALGNMmog8LyKjyxd7CQgDPheRzSIy5yKrc5t1BxrQK2oPEqA3PXFG55va0CggmwU/BNqOopS6QkGuWIkxZh4w74LXnq3weKgrxrlaZ7LPkFbUiht7LbcZwydIgJDUKp2vMzpRUlRCUIhLvoWUUh7gF1fabvx8Dw4C6XWtzpDpCiN+FsQpU5d103fYjqKUugJ+UfjrFpbNkNnz583tBvERQ6e2J4BS5s84bjuKUuoK+Efhb6lJ86CD1O/g2lM9/VW9FnXpWyeN+Rvr246ilLoC/lH4R5rSq9HByy+oqmxknxNsOJtA9lYrJ10ppa6Czxf+0dRc9pfG0quLzv/iSiPvLjs9c8Hruy0nUUpVlc8X/vpZewHoNbye5SS+pcvPy07PnLdQT89Uqrrw+cJfl3yWAErpfrPOkOlKEiCMbJ3OoqwEzp89bzuOUqoKfL/wt4fRMSSD0Pp6Sqar3TC2BnlE6OyZSlUTPn3VjHEY1p9oybg2qUBb23F8zpCpCdT4azHzPj7JwKm20yhPKDpVxO4fDrBz5TF2binm0JEAjufX4HhBCIXn/29iwuDAUurWKqJe6Hli6pUQ11yIaxtCy8RI2gyLIzhMb6Rjg08XfmbyAU6YOHom2k7im8JjwxlQbyPztjTmRdthlFuUFJWwbNo2Fn+Vx7LUeqw73Y5i2gBtEBzEyHGiapwmKuQMESFF/DRxybnSQA6ejmDrsTCOZkZRtKHWv9cZxHna1kynU4NcenUrpveIKLrfEk9I3RArn9Gf+HThr/v6EBBHr1ENbEfxWSP7n+Z/vu3OwbWHadq7se04ygWMw7DsjS18+s/TfLkzgWOmG4GUkBi6k0d6rqZH32Da9Y8i/rqm1I6OAS59fYtxGHJ35LJvXQ7p60+Rtvk8qZm1WHkojpkHmsBsqHF/Mb3DtzCo00kGjw7nmskJ+gPADcQYYztDpRITE01KSopT63i8RzJvb+zN6YIgatTWefDdYce3e0gY1Yp/TlzG5BkDbcdRTig8Ucgnj6fw95kNST0XT20KGNV0C7dMCGDYYx0Jaxjm8jGPbD7K2s/2sWpxIUu3R7OhoB2lBBFCIYOi0ri+/xlG3teMtkktdOLDKhKRDcaYSvdr+HThD4jYgsMIK093dlEqdSHjMLSsmUWnqMPMye5tO466CudOn+Ofv1zDH79OINfE0DlkF4/cmsOtL/WgdnRtj2bJP5zP8nd3suibsyzcHsvO4lYAtAzaz8867GPUbXUYNKWTbsBdgl8WfklRCeG1ipncdT1/3zTIhcnUhaZ2Xsr723py/LhQK7LW5f+B8grGYZj5yGqe+Ucse0uacW3dTTz7O2HQw128Zmt6/8os5r2ZyXc/hrA4pxNF1CKCPEbGpTJmjDDiVx0Jjw23HdOrXKrwffa0zO3flt3SsGdfnz5M4RVGTQijkNr8+Hqq7Siqivb8uJ+h0Zu47c1+hAcVsuCPKSw+3pXBj3b1mrIHiOsfywOfDuTbo704nmuY/fRabmqzlcUH2nDr6/2IaVqTkfXX887ty3Sajyrw2cJf/63e0tBTBj3UkTDymTur0HYUdRmlxaX8bVQynYbEsP5ka96esIyN+fEMfybRq4q+MrWjazP6hd68t2sAh4siWfHWVqb2WM2uk/W57+OBNO4STf/wrbx0QzK7F+61Hdcr+ewunfvaL2PWrs6cKInw+m9kXzC+yRrWZsdx8HxD/Xp7qaOpuUwceJDFJ7szqsFa3prbjNie1f+G9MZhSP06na+nHebrNQ3ZXNgOgPbBexjT/SCjfhFF718mEBjsH9OA+OUunfX7Y0isp7c09JRRI0o45GjEpk932o6iKrHklU107eJg5cn2vHvncmYf7uUTZQ9l03x0Gt+GZ38czKaz7di3IovXxi2lYe3TvLTmGvrf34lGISf4ResVfP74ak7tz7Md2RqfLPzCE4VsK2xNr3b5tqP4jZGPtUVwMPe9o7ajqAqMw/Dq2GSGPtGZukFnWPfFQe6ZPsCnN4Ti+sfy8JeD+PFkN47tK+DTqasY2mw3czI7cPOrfYluHsrAiC28MCyZlA+34yhx2I7sMT5Z+Ju/yKCEGvQcqGeMeEr9DjH0CUtj7jq9KYq3KCkqYUqXZTw+ezBjG69n/f4GdBrfxnYsj6obF8Gtr/fjk339yTlbhxVvbeXJvis4cz6Y3/4wmJ6TEqgffJKfx67m7QnL2DkvE+Pwzt3cruCTp7CsX3gCgJ7jm1lO4l9G9TvO04sGc3hjNo27N7Qdx6/lH87nlq47mZ87iCd7JfPnlQMJCPLJ7bsqCwoJov8Dnen/APyJsmMa30/bzeIfHCze25IvZjaBmVBfcrmm0R6u6VlE359F0+3meGqG17Qd3yVcctBWRJKA14BA4D1jzIsXvF8T+BDoARwHbjHG7LvUOp05aHtHyxX8uL8Vh0p9Yx9ldZH6dTqdxsXzzh3LuPdDverWlpN7TzGiUxYpBe14a+IqvQK6CozDkJl8gCXTD7B8hbDsQBz7SpoCEMw5uoWm06PFcXokBtA9qT7tRzT32h8Cbr3wSkQCgd3AMCALWA9MMMZsr7DMg0BnY8z9InIrcKMx5pZLrdeZwm8bvJf2UTl8c0Sv/PSkn6667Rh5hLlHe9mO45dydxzj+h7H2F7Yglm/2cyYP+n/A1frUMoR1s7az5rkItbursumvJbkU3aRVxDnaVNzP53q59C+VTFtOwXTtm8krQc1oU7jOlZzX6rwXbFLpxeQYYzJLB9sJjAGqDhJ+hjgufLHXwBviogYN5wTemp/HrvPt+DOjvtdvWp1GRIgjOmYyT829+ZM9hm3zL2iLi57aw5Dep0m81wcc/64jeHPaNk7o0liI8YlNmJc+XNHiYM9S/ax8dvDbNtQzNaMWqw7Esusg7GY5AB4o2y5aDlGq9rZNKuXT9MG52jaVGjcvAYNW4bSoHUdYtrUo25chJVdbK4o/CZAxTuEZwEXfqf9exljTImI5AFRwLGKC4nIZGAyQLNmV7///eWfJTP0Dt2dY8ONv4jgtUdDWPDSam76W1/bcfzG8fQTDO2Vx/5zjZj/6i4GP6pzgrtaQFAA8cOaEz+sORV3TxSeKCRjyUF2rTrOnu3nyNwXQGZOKFuONuTbrAYUbvjv+YgCKCVSjlEvKJ+IGoVE1CyiTkgxoTVLCQ0ppU1rB0/MHezyz+CKXTo3AUnGmHvKn98B9DbGTKmwTGr5Mlnlz/eUL3OssnWCayZPU55XUlRCw9p5JMXtZMbe/rbj+IXTWacZ2u4gWwtaMf9vO7j28W62I6lyxmE4seckR1KPk52eT/beQo5ll3D8mOHYiQBOnQkir7AGpwpDOFNSk4KSmhQ4atG57kEWHe9xVWO6e5fOIaBpheex5a9VtkyWiAQBEZQdvFU+JigkiNGtt/NVemeKzxTrnY3crPBEIaM6ZrKpoANf/XYT1z6ux068iQQIUfGRRMVH0vGK/qV7Tm92xU6k9UC8iLQQkWDgVmDOBcvMASaVP74J+NEd+++Vd7jxlprkEUHyG9tsR/FppcWlTOiwleV5nflo6npG/UHLXl2a04VvjCkBpgALgR3ALGNMmog8LyKjyxd7H4gSkQzgceApZ8dV3mvoY50I5QxfzyiwHcWnPdF7BbOze/Pa+OXc+no/23FUNeCzk6cpu26KXc2qIy3IOlff7y/4cYfXxi3l0a8H8Vj3ZF7ZMNh2HOVF/HLyNGXXjaMdHHE0ZN2/tl9+YXVF5jyzlse+HsCNjdbw0uoBtuOoakQLX7nFDb/uSBDn+frdi56Ipa5C2uwMJv4pgR61dzJja2e/mfJXuYYWvnKLunERXBe1hS83tvDpyag86cSek4y5qQahAYV8s7Sex+83q6o/LXzlNj+/oZA9JXFs/myX7SjVXmlxKRN67+FASSO+eusoTRL1wkJ15bTwldvc+NsOBFLCrNezbUep9p4ZuJxFxxOZdsca+t3XyXYcVU1p4Su3iYqPZEjUZmal6G4dZ8x5Zi1/WTuYye10FlLlHC185VY3/6yQzJI4Nn6itz68GnuXHWTSn9vSvdYOXlutF1Yp52jhK7ca+0wHgjjPrDf01odXquhUET8fkY8xwufzQgmpG2I7kqrmtPCVW0XFRzI0ejOzNrbS3TpX6IkBa9lwNoEPn95Jy8F69zblPC185XY3jypiX0lTUj7aYTtKtfHVk2t4K3UQT/RIZvQLOq+9cg0tfOV2Y/9fJ2pQzKw3c2xHqRYOrD7E3S+3I7H2dv6UrHPkKNfRwlduV69FXYbFbGHWptY4Shy243i1kqISbk86RokJ5NM5oTq9tHIpLXzlEd0ttEAAAA1vSURBVLfdVMyB0liWT9tqO4pXe2HECpaf7sLb92+l9ZA423GUj9HCVx4x9rmuhHKGj97Ktx3Fa61+ZxvPJw/g9hYruP1tvVuYcj0tfOURofVDGd9yM5/v7kzhiULbcbxO/uF8bn8onGZBh5m2rLPtOMpHaeErj7njvtqcJoK5f9hkO4rXeWTwZvaVxPLhqycIjw23HUf5KC185THXPtqFxgFH+OgTV9xK2Xd8+avVfJA+gKf6LWfAlC624ygfpoWvPCYwOJCJPXaxIKcbuTt0nnyAI5uPMvlvbUmsvZ3nvtf99sq9tPCVR93x68aUUIPPnk2zHcU64zDcc/0BzppafPR5CDVq17AdSfk4LXzlUZ3Gt6FLyC4+nB9tO4p17965nHm5Pfnr+HW0G9nSdhzlB5wqfBGJFJHvRSS9/O96lSzTVURWi0iaiGwVkVucGVNVf5OGH2F9QQe2fbnbdhRr9vy4n8c/7s6Qeht5aKbel1Z5hrNb+E8Bi40x8cDi8ucXOgvcaYzpACQBfxeRuk6Oq6qxO1/qTDDnePf5I7ajWFFaXMqdY/MIopQP5jckIEh/0Vae4ex32hhgevnj6cDYCxcwxuw2xqSXPz4M5AAxTo6rqrGo+Ehuikvho21dOHvsrO04HvfymOWsyu/MG/dto2nvxrbjKD/ibOE3MMb8tJmWDTS41MIi0gsIBvZc5P3JIpIiIim5ublORlPebPJjoZwydfni6Y22o3jUti938+yCvoxrvIbb39KzcpRniTGXnqNcRH4AGlby1jPAdGNM3QrLnjTG/Nd+/PL3GgHJwCRjzJrLBUtMTDQpKSmXW0xVU8ZhaBeyj5iQfFac9o8rS4vPFNMrZi9HzkWSmibEtNcD18r1RGSDMSaxsvcuu4VvjBlqjOlYyZ/ZwNHyIv+p0Cud/1ZEwoHvgGeqUvbK90mAMHn4flbmdyZtdobtOB7xfNIqthS15Z2nMrXslRXO7tKZA0wqfzwJmH3hAiISDHwNfGiM+cLJ8ZQPmfRyp7KDt88dsh3F7da8l8qfVw5gUqsVjPmT3tBE2eFs4b8IDBORdGBo+XNEJFFE3itf5mZgIPALEdlc/qerk+MqHxDdNopxzTYwfUsXCnIKbMdxm4KcAu58MJTYwCO89mMn23GUH3Oq8I0xx40xQ4wx8eW7fk6Uv55ijLmn/PEMY0wNY0zXCn82uyK8qv6mPBXGKVOXDx/ZYDuK2zw5JIX08y34119ziWgWYTuO8mN6ArCyqt99negVmsqrXzb1ybthLXwhhbdSB/FY92Sufbyb7TjKz2nhK6skQHj87tOkn2/Bd7/3rbOyTuw5yV3PNqF98B5e+F732yv7tPCVdeP/0otmgVn87Q3fuX+rcRjuH7yTHEc0H71fTK3IWrYjKaWFr+wLCgni4REZLM3ryoYZO2zHcYmP7l/J51l9ef76lfS4vb3tOEoBWvjKS9wzrRth5PPq/zthO4rT9i47yJR3OzMgfAtPztWJ0ZT30MJXXiGiWQT3dNvIZ/t6sW9Flu04V62kqITbf3YSwfDRvCgCgwNtR1Lq37Twldd44p9tCMDBH+/eazvKVftj0gpW5XfmrQdSiesfazuOUv9BC195jdiejbiv8xr+tbsvGYv3245zxZa+tpk/LB3AnS1XMFEnRlNeSAtfeZXfTG9HDc7zh8kHbUe5Isd2HWfi4w1oVeMA01bqheTKO2nhK6/SqGsDHuqxlhmZfdk5L9N2nCoxDsNdgzLIdUQy84MiwhqG2Y6kVKW08JXXeXJ6B2pRyO8fyLYdpUpeGbOUuUd789K4NXSfqKdgKu+lha+8Tv0OMUztk8JnB/qw6dOdtuNc0tLXNvPrb69hXOM1TP18oO04Sl2SFr7ySk9+0pVoOc6U+4q9do6dQylHuPmxJrQOPsAHaxOQALEdSalL0sJXXqlei7r85Rc7WZXfmQ/vX2U7zn8pPlPMzdflUmBq8dXnDsJjw21HUuqytPCV15r0Tn/6hm3jyf9ty6n9ebbj/JtxGKb0XMOq/M68//BWEka3th1JqSrRwldeKyAogGnv1OC4ieT/jfKeWyi8MmYp7+4cyG/6JnPLa/1sx1GqyrTwlVfrNqEdD3RawVvbrmHdB2m24zD76bX86tuB3NRkNX9cpgdpVfWiha+83h/ndqVJYDYTJodxOuu0tRwbZuzgtj93JDF0B9M3dyEgSP/3UdWLfscqr1c3LoKP/36MfSWxPHDNNozDeDxD2uwMht9Zn+jAk8xeEU3t6Noez6CUs7TwVbUwYEoXfnftcj7Z358P71vp0bH3/LifYePCqCElLF5YSqOuDTw6vlKu4lThi0ikiHwvIunlf9e7xLLhIpIlIm86M6byX88sGMCgiM089F5X0mZneGTMA6sPMeT6QIpNDX74Kp/WQ+I8Mq5S7uDsFv5TwGJjTDywuPz5xfwBWObkeMqPBQYHMuP7BtQJKOD6caHsXebeCdbSZmfQf4BwsrQOCz/MocMYPf1SVW/OFv4YYHr54+nA2MoWEpEeQANgkZPjKT8X27MRi2blUWhCGDbEQfbWHLeMs/zNLVxzYzSlJoBln2XrbQqVT3C28BsYY46UP86mrNT/g4gEAH8D/udyKxORySKSIiIpubm5TkZTvqrT+DbMe+cQR0qiub73KY6nu/a2iJ88tJJhU9vSoMYJVi0tocvNbV26fqVsuWzhi8gPIpJayZ8xFZczxhigstMnHgTmGWMue986Y8w7xphEY0xiTExMlT+E8j997unIN3/eya6iOBITCtj4sfM3P88/nM+kVmU3L+kZvouVqXVpfo3etUr5jssWvjFmqDGmYyV/ZgNHRaQRQPnflf1+3ReYIiL7gJeBO0XkRRd+BuWnhj3Vg2XvpVNiAul3ews+uGv5Va9r2Rtb6BZ3ghmZfXl2YDJLjnYgKj7ShWmVss/ZXTpzgEnljycBsy9cwBgz0RjTzBjTnLLdOh8aYy51cFepKut9d0c2bgvmmnrbueuDAYyIWc/qd7ZV+d+v+yCNpOgUBj3chfMmkKXT0vj90sEEhQS5MbVSdjhb+C8Cw0QkHRha/hwRSRSR95wNp1RVxLSPZmF2F/46MpmU4y3od18nro/awEf3r2D3wr3/caGWcRjSZmfw0g3JDKq7md53dSDlRAteuiGZHdmRXPNgZ4ufRCn3krJd794nMTHRpKSk2I6hqpkz2Wf4xz0pvDw/gaOO+gDUk5PUCzzNmdLa5JtQCim7SrZzyC5uHXiEKR/0oE7jOjZjK+UyIrLBGJNY6Xta+MoXlRaXsuO7TNbOzmbdWsPZcwGEhjgIre2gXUIAI6bGE9uzke2YSrmcFr5SSvmJSxW+zqWjlFJ+QgtfKaX8hBa+Ukr5CS18pZTyE1r4SinlJ7TwlVLKT2jhK6WUn9DCV0opP+G1F16JSC6w34lVRAPHXBSnuvC3z+xvnxf0M/sLZz5znDGm0vnlvbbwnSUiKRe72sxX+dtn9rfPC/qZ/YW7PrPu0lFKKT+hha+UUn7Clwv/HdsBLPC3z+xvnxf0M/sLt3xmn92Hr5RS6j/58ha+UkqpCrTwlVLKT/hc4YtIkojsEpEMEfH5m6WLSFMRWSIi20UkTUQesZ3JU0QkUEQ2ici3trN4gojUFZEvRGSniOwQkb62M7mbiDxW/n2dKiKfikiI7UyuJiL/KyI5IpJa4bVIEfleRNLL/67nirF8qvBFJBCYBowAEoAJIpJgN5XblQBPGGMSgD7AQ37wmX/yCLDDdggPeg1YYIxpB3TBxz+7iDQBHgYSjTEdgUDgVrup3OJfQNIFrz0FLDbGxAOLy587zacKH+gFZBhjMo0xxcBMYIzlTG5ljDlijNlY/jifshJoYjeV+4lILHAD8J7tLJ4gIhHAQOB9AGNMsTHmlN1UHhEE1BKRIKA2cNhyHpczxiwDTlzw8hhgevnj6cBYV4zla4XfBDhY4XkWflB+PxGR5kA3YK3dJB7xd+BJwGE7iIe0AHKBD8p3Y70nIqG2Q7mTMeYQ8DJwADgC5BljFtlN5TENjDFHyh9nAw1csVJfK3y/JSJhwJfAo8aY07bzuJOI/AzIMcZssJ3Fg4KA7sDbxphuQAEu+jXfW5Xvtx5D2Q+7xkCoiNxuN5XnmbJz511y/ryvFf4hoGmF57Hlr/k0EalBWdl/bIz5ynYeD+gPjBaRfZTttrtORGbYjeR2WUCWMean396+oOwHgC8bCuw1xuQaY84DXwH9LGfylKMi0gig/O8cV6zU1wp/PRAvIi1EJJiyAzxzLGdyKxERyvbr7jDGvGI7jycYY35jjIk1xjSn7L/xj8YYn97yM8ZkAwdFpG35S0OA7RYjecIBoI+I1C7/Ph+Cjx+ormAOMKn88SRgtitWGuSKlXgLY0yJiEwBFlJ2RP9/jTFplmO5W3/gDmCbiGwuf+1pY8w8i5mUe0wFPi7fmMkEfmk5j1sZY9aKyBfARsrORtuED06zICKfAoOBaBHJAn4HvAjMEpG7KZsm/maXjKVTKyillH/wtV06SimlLkILXyml/IQWvlJK+QktfKWU8hNa+Eop5Se08JVSyk9o4SullJ/4/6uezNnrlrZfAAAAAElFTkSuQmCC\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "x = np.linspace(-0, 10, 100)\n",
    "y0 = func0(x)\n",
    "y =func(x)\n",
    "plt.plot(x,y0,color='red');\n",
    "plt.plot(x,y,color='blue');\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "id": "17bcc35f-32da-4b92-bbea-2f937f204985",
   "metadata": {},
   "outputs": [],
   "source": [
    "xpoints1 = np.random.rand(1,n)\n",
    "ypoints =  np.random.rand(1,n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "id": "8ab4ac05-2f53-49ef-a226-f1ff07371f97",
   "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.plot(xpoints1,ypoints,'o',color='black');\n",
    "plt.plot(x,y,color='red');"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "id": "d4e64e30-1e8a-42c0-955d-7c4e056085cd",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "50"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "ychecks1 = func(xpoints1)\n",
    "n1=np.count_nonzero(ypoints<=ychecks1)\n",
    "n1"
   ]
  }
 ],
 "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": "3.6.9"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 5
}