RepozytoriumzprojektemPython/stopy/Ćwiczenia_10.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "id": "4cc6c96f",
   "metadata": {},
   "source": [
    "# Ćwiczenia 10"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1276423e",
   "metadata": {},
   "source": [
    "***TEMAT:*** całki podwójne"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e3d9e3c2",
   "metadata": {},
   "source": [
    "## Całka podwójna po prostokącie"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1bcce97a",
   "metadata": {},
   "source": [
    "### Definicja: sumy dolne, sumy górne, całka"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "64ff5ed6",
   "metadata": {},
   "source": [
    "#### Przykład:\n",
    "\n",
    "Korzystając z definicji obliczymy \n",
    "\n",
    "$$\\iint_{[0,1]\\times [0,1]}xydxdy.$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "cf058d53",
   "metadata": {},
   "source": [
    "__UWAGA:__ Sumy dolne, sumy górne, całkę dolną, całkę górną i całkę po prostokącie definiuje się w identycznycny sposób jak odpowiedniki tych pojęć dla przedziału. "
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1fdd0220",
   "metadata": {},
   "source": [
    "1. Dzielimy prostokąt $[0,1]\\times [0,1]$ na $n^2$ mniejszych prostokątów o równych polach \n",
    "w następujący sposób (rysunek dla $n=5$), podział ten nazywamy podziałem $P_n$."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "29b88b52",
   "metadata": {},
   "outputs": [
    {
     "data": {
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      "text/plain": [
       "<Figure size 600x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import matplotlib.pyplot as plt\n",
    "import numpy as np\n",
    "\n",
    "# Define the square grid [0,1] x [0,1] divided into 25 equal squares\n",
    "n = 5  # 5x5 grid\n",
    "x = np.linspace(0, 1, n+1)\n",
    "y = np.linspace(0, 1, n+1)\n",
    "\n",
    "# Plot the grid\n",
    "fig, ax = plt.subplots(figsize=(6, 6))\n",
    "for i in x:\n",
    "    ax.plot([i, i], [0, 1], color=\"black\", linewidth=0.8)  # Vertical lines\n",
    "for j in y:\n",
    "    ax.plot([0, 1], [j, j], color=\"black\", linewidth=0.8)  # Horizontal lines\n",
    "\n",
    "# Set aspect ratio and limits\n",
    "ax.set_aspect('equal', adjustable='box')\n",
    "ax.set_xlim(0, 1)\n",
    "ax.set_ylim(0, 1)\n",
    "ax.set_title(\"Square [0,1] x [0,1] Divided into 25 Smaller Squares\")\n",
    "ax.set_xlabel(\"x-axis\")\n",
    "ax.set_ylabel(\"y-axis\")\n",
    "plt.grid(False)  # Disable the background grid\n",
    "\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0ce9ffdb",
   "metadata": {},
   "source": [
    "2. Dla $1\\leq  i\\leq n$ oraz $1\\leq j\\leq n$ niech \n",
    "\n",
    "$$\n",
    "P_{i,j}=\\left[\\frac{i-1}{n},\\frac{i}{n}\\right]\\times\\left[\\frac{j-1}{n},\\frac{j}{n}\\right].\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3782244e",
   "metadata": {},
   "source": [
    "3. Dla $1\\leq i\\leq n$ oraz $1\\leq j\\leq n$ mamy\n",
    "\n",
    "$$\n",
    "\\inf_{(x,y)\\in P_{i,j}} f(x,y)=\\inf_{(x,y)\\in P_{i,j}} xy=\\frac{(i-1)(j-1)}{n^2}\n",
    "$$\n",
    "\n",
    "oraz\n",
    "\n",
    "$$\n",
    "\\sup_{(x,y)\\in P_{i,j}} f(x,y)=\\sup_{(x,y)\\in P_{i,j}} xy=\\frac{ij}{n^2}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1251d3c1",
   "metadata": {},
   "source": [
    "4. Mamy (korzystając ze wzoru na sumę ciągu arytmetycznego)\n",
    "\n",
    "$$\n",
    "L(f,P_n)=\\sum_{i=1}^n\\sum_{j=1}^n \\frac{1}{n^2}\\cdot \\inf_{(x,y)\\in P_{i,j}}f(x,y)=\n",
    "\\sum_{i=1}^n\\sum_{j=1}^n \\frac{(i-1)(j-1)}{n^4}=\\frac{(n-1)^2}{4n^2}\\xrightarrow{n\\to\\infty}\\frac{1}{4}\n",
    "$$\n",
    "\n",
    "oraz \n",
    "\n",
    "$$\n",
    "U(f,P_n)=\\sum_{i=1}^n\\sum_{j=1}^n \\frac{1}{n^2}\\cdot \\sup_{(x,y)\\in P_{i,j}}f(x,y)=\n",
    "\\sum_{i=1}^n\\sum_{j=1}^n \\frac{ij}{n^4}=\\frac{(n+1)^2}{4n^2}\\xrightarrow{n\\to\\infty}\\frac{1}{4}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a3b52dbe",
   "metadata": {},
   "source": [
    "5. Rachunki powyżej implikują, że \n",
    "\n",
    "$$\n",
    "\\iint_{[0,1]\\times [0,1]}xydxdy=\\frac{1}{4}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "543cca36",
   "metadata": {},
   "source": [
    "### Twierdzenie Fubiniego dla prostokąta.\n",
    "\n",
    "__Twierdzenie.__  \n",
    "Jeśli $f(x,y)$ jest funkcją ciągłą na prostokącie $R=[a,b] \\times [c,d]$, to całkę podwójną można obliczyć poprzez całki iterowane:\n",
    "\n",
    "$$ {\\color{red}{\n",
    "\\iint_R f(x,y) \\, dxdy = \\int_a^b \\int_c^d f(x,y) \\, dy \\, dx =\\int_c^d \\int_a^b f(x,y) \\, dx \\, dy. }}\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "70e22e12",
   "metadata": {},
   "source": [
    "#### Zadanie 1\n",
    "\n",
    "Obliczymy \n",
    "\n",
    "$$\n",
    "\\iint_{[0,1]\\times [0,1]} xydxdy.\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aa5e2fd0",
   "metadata": {},
   "source": [
    "##### Rozwiązanie:\n",
    "Z twierdzenia Fubiniego wynika, że \n",
    "\n",
    "$$\\iint_{[0,1]\\times [0,1]} xydxdy=\\int_0^1\\int_0^1xydydx=\n",
    "\\int_0^1\\frac{xy^2}{2}\\biggr|_{y=0}^{y=1}dx=\n",
    "\\int_0^1 \\frac{x}{2}dx=\\frac{x^2}{4}\\biggr|_{x=0}^{x=1}=\\frac{1}{4}. \n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "540a109a",
   "metadata": {},
   "source": [
    "#### Zadanie 2\n",
    "\n",
    "Obliczymy \n",
    "\n",
    "$$\n",
    "\\iint_{[1,4]\\times [2,3]} (2x+4y)dxdy.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "6ec05c7d",
   "metadata": {},
   "source": [
    "##### Rozwiązanie:\n",
    "__I sposób__\n",
    "$$\n",
    "\\iint_{[1,4]\\times [2,3]} (2x+4y)dxdy=\\int_1^4\\int_2^3(2x+4y)dydx=\n",
    "\\int_1^4 2xy+2y^2\\biggr|_{y=2}^{y=3}dx=\\int_1^42x+10dx=\n",
    "x^2+10x\\biggr|_{x=1}^{x=4}=45.\n",
    "$$\n",
    "\n",
    "__II sposób__\n",
    "$$\n",
    "\\iint_{[1,4]\\times [2,3]} (2x+4y)dxdy=\\int_2^3\\int_1^4(2x+4y)dxdy=\n",
    "\\int_2^3 x^2+4yx\\biggr|_{x=1}^{x=4}dy=\\int_2^3 15+12y dy=\n",
    "15y +6y^2\\biggr|_{y=2}^{y=3}=45.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c5c2e7a8",
   "metadata": {},
   "source": [
    "### UWAGA!!!\n",
    "\n",
    "Czasami wybór kolejności całkowania ma istotne  znaczenie!"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "34ca0028",
   "metadata": {},
   "source": [
    "#### Zadanie 3\n",
    "\n",
    "Obliczymy \n",
    "\n",
    "$$\n",
    "\\iint_{[-1,1]\\times [0,2]} xe^{y^2}dxdy.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "bdc0512f",
   "metadata": {},
   "source": [
    "##### Rozwiązanie:\n",
    "Gdybyśmy najpierw chcieli całkować \"po y\", to musielibyśmy zmierzyć się z całką $\\int e^{y^2}dy\\ldots$.\n",
    "Dlatego robimy tak:\n",
    "\n",
    "$$\n",
    "\\iint_{[-1,1]\\times [0,2]} xe^{y^2}dxdy=\\int_0^2\\int_{-1}^1 xe^{y^2}dxdy=\n",
    "\\int_0^2\\frac{x^2e^{y^2}}{2}\\biggr|_{x=-1}^{x=1}=0.\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "5eaec7e6",
   "metadata": {},
   "source": [
    "### Zadanie domowe (z wykładu):\n",
    "\n",
    "1. Obliczyć $ \\int_1^4 \\int_0^2 \\left(6x^2 y - 2x\\right) \\, dy \\, dx$.\n",
    "2. Obliczyć $\\int_1^3 \\int_1^5 \\dfrac{\\ln y}{xy} \\, dy \\, dx$.\n",
    "3. Obliczyć $ \\int_0^1 \\int_1^2 \\left( x + e^{-y} \\right) \\, dx \\, dy$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "b08fa672",
   "metadata": {},
   "source": [
    "## Zastosowanie"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "3c3c8fba",
   "metadata": {},
   "source": [
    "### Objętość\n",
    "\n",
    "Niech $f$ oraz $g$ będą funkcjami ciągłymi na prostokącie $[a,b]\\times [c,d]$ takimi , że \n",
    "\n",
    "$$\n",
    "f(x,y)\\leq g(x,y) \\quad\\text{dla}\\quad (x,y)\\in [a,b]\\times [c,d].\n",
    "$$\n",
    "\n",
    "Objętość bryły\n",
    "\n",
    "$$\n",
    "E=\\{(x,y,z): (x,y)\\in [a,b]\\times [c,d], f(x,y)\\leq z\\leq g(x,y)\\}\n",
    "$$\n",
    "\n",
    "wyraża się wzorem\n",
    "\n",
    "$$\n",
    "\\iint_{[a,b]\\times [c,d]}(g(x,y)-(f(x,y))dxdy.\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "4b6ddb20",
   "metadata": {},
   "source": [
    "#### Zadanie 4"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "0f22cdf5",
   "metadata": {},
   "source": [
    "Obliczymy objętość prostopadłościanu $[0,1]\\times [0,1]$ \"ściętego z góry\" płaszczyzną $x+y+z=3$ a z dołu płaszczyzną $z=0$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "id": "28e06198",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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
      "text/plain": [
       "<Figure size 800x800 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "from mpl_toolkits.mplot3d import Axes3D\n",
    "\n",
    "# Define the range for x and y\n",
    "x = np.linspace(0, 1, 30)\n",
    "y = np.linspace(0, 1, 30)\n",
    "x, y = np.meshgrid(x, y)\n",
    "\n",
    "# Define the bounds for z\n",
    "z_upper = 3 - x - y\n",
    "z_lower = np.zeros_like(z_upper)\n",
    "\n",
    "# Create a 3D plot\n",
    "fig = plt.figure(figsize=(8, 8))\n",
    "ax = fig.add_subplot(111, projection='3d')\n",
    "\n",
    "# Plot the surfaces\n",
    "ax.plot_surface(x, y, z_upper, alpha=0.7, color='blue', edgecolor='k', label=\"z = 3 - x - y\")\n",
    "ax.plot_surface(x, y, z_lower, alpha=0.3, color='cyan', edgecolor='k', label=\"z = 0\")\n",
    "\n",
    "# Set labels and limits\n",
    "ax.set_xlabel('X-axis')\n",
    "ax.set_ylabel('Y-axis')\n",
    "ax.set_zlabel('Z-axis')\n",
    "ax.set_xlim(0, 1)\n",
    "ax.set_ylim(0, 1)\n",
    "ax.set_zlim(0, 3)\n",
    "\n",
    "# Title\n",
    "ax.set_title('Visualization of the set: x ∈ [0,1], y ∈ [0,1], 0 ≤ z ≤ 3-x-y')\n",
    "\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "e7f0cc7a",
   "metadata": {},
   "source": [
    "##### Rozwiązanie:\n",
    "\n",
    "Musimy obliczyć \n",
    "\n",
    "$$\n",
    "\\iint_{[0,1]\\times [0,1]}(3-x-y)dxdy.\n",
    "$$\n",
    "\n",
    "Mamy\n",
    "\n",
    "$$\n",
    "\\iint_{[0,1]\\times [0,1]}(3-x-y)dxdy=\\int_0^1\\int_0^1(3-x-y)dydx=\n",
    "\\int_0^1 3y-xy-\\frac{y^2}{2}\\biggr|_{y=0}^{y=1}dx=\n",
    "\\int_0^1 \\frac{5}{2}-xdx=\\frac{5}{2}x-\\frac{x^2}{2}\\biggr|_{x=0}^{x=1}=2.\n",
    "$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c50d6b0a",
   "metadata": {},
   "source": [
    "## Całka podwójna po obszarze"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "ff3d0827",
   "metadata": {},
   "source": [
    "### Przypomnienie z wykładu\n",
    "\n",
    "Niech $f$ będzie funkcją określoną i ograniczoną na obszarze ograniczonym $D\\subset\\mathbb{R}^2$ oraz niech $R$ będzie dowolnym prostokątem zawierającym obszar $D.$ Ponadto niech funkcja $f^{\\ast}$ będzie rozszerzeniem funkcji $f$ z $D$ na $R$ określonym wzorem:\n",
    "\n",
    "$$\n",
    "f^{\\ast}(x,y)=\\begin{cases}f(x,y)\\quad {\\rm dla}\\quad (x,y)\\in D\\\\ 0\\;\\,\\qquad\\quad {\\rm dla}\\quad (x,y)\\in R\\setminus D.\\end{cases}\n",
    "$$\n",
    "__Całkę podwójną__ funkcji $f$ po obszarze $D$ definiujemy wzorem:\n",
    "\n",
    "$$\n",
    "\\displaystyle\\iint\\limits_Df(x,y) \\; dxdy=\\iint\\limits_Rf^{\\ast}(x,y) \\; dxdy\n",
    "$$\n",
    "\n",
    " o ile całka poprawej stronie istnieje. Mówimy wtedy, że funkcja $f$ jest całkowalna na obszarze $D.$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "aa7bd785",
   "metadata": {},
   "source": [
    "### Zbiory normalne\n",
    "\n",
    "__Obszar normalny__ względem osi $OX$ to zbiór punktów $(x,y)$ spełniających warunek\n",
    "\n",
    "$$\n",
    "\\begin{cases}a\\leq x\\leq b\\\\f(x)\\leq y\\leq g(x),\\end{cases}\n",
    "$$\n",
    "\n",
    "a obszar normalny względem osi $OY$ to następujący zbiór punktów\n",
    "\n",
    "$$\n",
    "\\begin{cases}c\\leq y\\leq d\\\\ f(y)\\leq x\\leq g(y),\\end{cases}\n",
    "$$\n",
    "gdzie funkcje $f$ oraz $g$ są ciągłe."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "25754b0c",
   "metadata": {},
   "source": [
    "#### Zadanie 5"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "16ef10ce",
   "metadata": {},
   "source": [
    "Czy poniższe zbiory są normalne względem którejś z osi:\n",
    "\n",
    "1. Zbiór $D$ to trójkąt o wierzchołkach w punktach $(0,0)$, $(1,0)$, $(1,4)$.\n",
    "2. Zbiór $D$ to koło ośrodku w punkcie $(0,0)$ i promieniu $2$.\n",
    "3. Zbiór $D$ to zbiór znajdujący się pomiędzy krzywą $y=x^2$ a prostą $y=1$.\n",
    "4. Zbiór $D$ to zbiór znajdujący się pomiędzy krzywą $x=y^2$ a prostą $x=1$.\n",
    "5. Zbiór $D$ to kwadrat o wierzchołkach w punktach $(-1,0)$, $(1,0)$, $(0,-1)$, $(0,1)$."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "1dcfd799",
   "metadata": {},
   "source": [
    "##### Rozwiązanie:"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "23b46f13",
   "metadata": {},
   "source": [
    "1. Mamy \n",
    "\n",
    "$$D=\\{(x,y): 0\\leq x\\leq 1, 0\\leq y\\leq 4x\\}$$\n",
    "\n",
    "oraz\n",
    "\n",
    "$$\n",
    "D=\\{(x,y): 0\\leq y\\leq 4, \\frac{y}{4}\\leq x\\leq 1\\}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "id": "401d4908",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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
      "text/plain": [
       "<Figure size 800x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Define the range for x\n",
    "x = np.linspace(0, 1, 100)\n",
    "\n",
    "# Define the boundary for y\n",
    "y = 4 * x\n",
    "\n",
    "# Create the plot\n",
    "plt.figure(figsize=(8, 6))\n",
    "plt.fill_between(x, 0, y, color=\"skyblue\", alpha=0.5, label=r\"$D = \\{(x,y): 0 \\leq x \\leq 1, 0 \\leq y \\leq 4x\\}$\")\n",
    "plt.plot(x, y, color=\"blue\", label=r\"$y = 4x$\")\n",
    "\n",
    "# Add labels, title, and legend\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"y\")\n",
    "plt.title(\"Region D: $0 \\leq x \\leq 1, 0 \\leq y \\leq 4x$\")\n",
    "plt.legend()\n",
    "plt.grid(True)\n",
    "\n",
    "# Show the plot\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "01365abd",
   "metadata": {},
   "source": [
    "2. Mamy\n",
    "$$D=\\{(x,y): -2\\leq x\\leq 2, -\\sqrt{4-x^2}\\leq y\\leq \\sqrt{4-x^2}\\}$$\n",
    "\n",
    "oraz\n",
    "\n",
    "$$\n",
    "D=\\{(x,y): -2\\leq y\\leq 2, -\\sqrt{4-y^2}\\leq x\\leq \\sqrt{4-y^2}\\}.\n",
    "$$"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "8617403f",
   "metadata": {},
   "source": [
    "3. Mamy\n",
    "$$D=\\{(x,y): -1\\leq x\\leq 1, x^2\\leq y\\leq 1\\}$$\n",
    "oraz \n",
    "$$D=\\{(x,y): 0\\leq y\\leq 1, -\\sqrt{y}\\leq x\\leq \\sqrt{y}\\}.$$\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "id": "71dd69d9",
   "metadata": {},
   "outputs": [
    {
     "data": {
      "image/png": "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
      "text/plain": [
       "<Figure size 800x600 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "import numpy as np\n",
    "import matplotlib.pyplot as plt\n",
    "\n",
    "# Define the range for x\n",
    "x = np.linspace(-1, 1, 100)\n",
    "\n",
    "# Define the boundary for y\n",
    "y_upper = np.ones_like(x)  # y = 1\n",
    "y_lower = x**2  # y = x^2\n",
    "\n",
    "# Create the plot\n",
    "plt.figure(figsize=(8, 6))\n",
    "plt.fill_between(x, y_lower, y_upper, color=\"skyblue\", alpha=0.5, label=r\"$D = \\{(x,y): -1 \\leq x \\leq 1, x^2 \\leq y \\leq 1\\}$\")\n",
    "plt.plot(x, y_upper, color=\"blue\", label=r\"$y = 1$\")\n",
    "plt.plot(x, y_lower, color=\"red\", label=r\"$y = x^2$\")\n",
    "\n",
    "# Add labels, title, and legend\n",
    "plt.xlabel(\"x\")\n",
    "plt.ylabel(\"y\")\n",
    "plt.title(\"Region D: $-1 \\leq x \\leq 1, x^2 \\leq y \\leq 1$\")\n",
    "plt.legend()\n",
    "plt.grid(True)\n",
    "\n",
    "# Show the plot\n",
    "plt.show()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "a61f2d80",
   "metadata": {},
   "source": [
    "4. Mamy\n",
    "$$D=\\{(x,y): -1\\leq y\\leq 1, y^2\\leq x\\leq 1\\}$$\n",
    "oraz \n",
    "$$D=\\{(x,y): 0\\leq x\\leq 1, -\\sqrt{x}\\leq y\\leq \\sqrt{x}\\}.$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "d845490d",
   "metadata": {},
   "source": [
    "5. Mamy\n",
    "$$D=\\{(x,y): -1\\leq x\\leq 1, |x|-1\\leq y\\leq 1-|x|\\}$$\n",
    "oraz \n",
    "$$D=\\{(x,y): -1\\leq y\\leq 1, |y|-1\\leq x\\leq 1-|y|\\}.$$\n"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "9a3bd752",
   "metadata": {},
   "source": [
    "#### Zadanie 6"
   ]
  },
  {
   "cell_type": "markdown",
   "id": "04fcfaac",
   "metadata": {},
   "source": [
    "Podaj przykład zbioru, który nie jest normalny względem żadnej z osi."
   ]
  },
  {
   "cell_type": "markdown",
   "id": "c3aa7415",
   "metadata": {},
   "source": [
    "##### Rozwiązanie:\n",
    "Przykładem takiego zbioru jest:\n",
    "\n",
    "$$D=\\{(x,y): 1\\leq x^2+y^2\\leq 4\\}.$$\n",
    "\n",
    "Zauważmy, że dzieląc ten zbiór prostą $y=0$ dostalibyśmy dwa zbiory normalne względem osi OX."
   ]
  }
 ],
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  "kernelspec": {
   "display_name": "Python 3 (ipykernel)",
   "language": "python",
   "name": "python3"
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    "version": 3
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   "file_extension": ".py",
   "mimetype": "text/x-python",
   "name": "python",
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   "pygments_lexer": "ipython3",
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  "toc": {
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   "title_cell": "Table of Contents",
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