Independent Work 3.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Numerical Integration"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<p> In this independent work, you are going to implement one of three numerical integration methods: Trapezoid, Midpoint, or Simpson's methods.</p>\n",
    "<p> To determine your method, your position in the class list modulo 3. \n",
    "    <ul>\n",
    "        <li>0: Trapezoid </li>\n",
    "        <li>1: Midpoint </li>\n",
    "        <li>2: Simpson </li>\n",
    "    </ul>\n",
    "</p>\n",
    "\n",
    "<p>For more information, check out the textbook. </p>\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Task 1: Linear Approximation\n",
    "<p>Implement a function that takes a function and a closed interval [a, b] as an input, and returns a numerical approximation of the integral of the function over the given interval. Moreover, try to draw the graph of the function and the area under the curve. </p>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sym\n",
    "import numpy as np\n",
    "import matplotlib.pyplot as aizada\n",
    "%matplotlib inline\n",
    "from sympy.abc import x"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "x**2\n"
     ]
    }
   ],
   "source": [
    "f = input()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'x**2'"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "f"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "fsym= sym.sympify(f)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "sympy.core.power.Pow"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(fsym)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [],
   "source": [
    "flam= sym.lambdify(x,fsym)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "flam(2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "0 2\n"
     ]
    }
   ],
   "source": [
    "a, b = input().split()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [],
   "source": [
    "a = int(a)\n",
    "b = int(b)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "int"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "type(a)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 12,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "4\n"
     ]
    }
   ],
   "source": [
    "n = input()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "metadata": {},
   "outputs": [],
   "source": [
    "n= int(n)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 14,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "4"
      ]
     },
     "execution_count": 14,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 15,
   "metadata": {},
   "outputs": [],
   "source": [
    "dx = (b-a)/n\n",
    "\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 16,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "0.5"
      ]
     },
     "execution_count": 16,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "dx"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 21,
   "metadata": {},
   "outputs": [],
   "source": [
    "xi= [a+i*dx for i in range(0,n+1)]"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "5"
      ]
     },
     "execution_count": 23,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "len(xi)\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 23,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "1.25\n",
      "1.5\n",
      "1.75\n",
      "2.0\n"
     ]
    }
   ],
   "source": [
    "for i in range(1, n+1):\n",
    "    print(xi[i])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 24,
   "metadata": {},
   "outputs": [],
   "source": [
    "sum1 = 0\n",
    "for i in range(1,n+1):\n",
    "    sum1 += flam(xi[i])*dx\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 26,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.71875"
      ]
     },
     "execution_count": 26,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "sum1"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 27,
   "metadata": {},
   "outputs": [],
   "source": [
    "x0 = np.linspace(0, 2, 200)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 28,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "[<matplotlib.lines.Line2D at 0x1f9e1414c10>]"
      ]
     },
     "execution_count": 28,
     "metadata": {},
     "output_type": "execute_result"
    },
    {
     "data": {
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\n",
      "text/plain": [
       "<Figure size 432x288 with 1 Axes>"
      ]
     },
     "metadata": {
      "needs_background": "light"
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "aizada.plot(x0, flam(x0))"
   ]
  }
 ],
 "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": "3.8.8"
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}