{"id":21413,"date":"2025-08-15T20:13:27","date_gmt":"2025-08-15T12:13:27","guid":{"rendered":"https:\/\/blog.iyatt.com\/?p=21413"},"modified":"2025-08-22T12:40:17","modified_gmt":"2025-08-22T04:40:17","slug":"%e6%98%9f%e5%bd%a2%e7%ba%bf","status":"publish","type":"post","link":"https:\/\/blog.iyatt.com\/?p=21413","title":{"rendered":"\u661f\u5f62\u7ebf"},"content":{"rendered":"<div id=\"ez-toc-container\" class=\"ez-toc-v2_0_82_2 ez-toc-wrap-center counter-hierarchy ez-toc-counter ez-toc-light-blue ez-toc-container-direction\">\n<div class=\"ez-toc-title-container\">\n<p class=\"ez-toc-title ez-toc-toggle\" style=\"cursor:pointer\">\u76ee\u5f55<\/p>\n<span class=\"ez-toc-title-toggle\"><a href=\"#\" class=\"ez-toc-pull-right ez-toc-btn ez-toc-btn-xs ez-toc-btn-default ez-toc-toggle\" aria-label=\"Toggle Table of Content\"><span class=\"ez-toc-js-icon-con\"><span class=\"\"><span class=\"eztoc-hide\" style=\"display:none;\">Toggle<\/span><span class=\"ez-toc-icon-toggle-span\"><svg 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><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-13\" href=\"https:\/\/blog.iyatt.com\/?p=21413\/#%E6%96%B9%E6%B3%95%E4%B8%80%EF%BC%9A%E7%9B%B4%E8%A7%92%E5%9D%90%E6%A0%87%E6%96%B9%E7%A8%8B-3\" >\u65b9\u6cd5\u4e00\uff1a\u76f4\u89d2\u5750\u6807\u65b9\u7a0b<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-14\" href=\"https:\/\/blog.iyatt.com\/?p=21413\/#%E6%96%B9%E6%B3%95%E4%BA%8C%EF%BC%9A%E5%8F%82%E6%95%B0%E6%96%B9%E7%A8%8B-3\" >\u65b9\u6cd5\u4e8c\uff1a\u53c2\u6570\u65b9\u7a0b<\/a><\/li><\/ul><\/li><li class='ez-toc-page-1 ez-toc-heading-level-2'><a class=\"ez-toc-link ez-toc-heading-15\" href=\"https:\/\/blog.iyatt.com\/?p=21413\/#%E6%97%8B%E8%BD%AC%E4%BD%93%E4%BE%A7%E9%9D%A2%E7%A7%AF%E8%AE%A1%E7%AE%97\" >\u65cb\u8f6c\u4f53\u4fa7\u9762\u79ef\u8ba1\u7b97<\/a><ul class='ez-toc-list-level-3' ><li class='ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-16\" href=\"https:\/\/blog.iyatt.com\/?p=21413\/#%E6%96%B9%E6%B3%95%E4%B8%80%EF%BC%9A%E7%9B%B4%E8%A7%92%E5%9D%90%E6%A0%87%E6%96%B9%E7%A8%8B-4\" >\u65b9\u6cd5\u4e00\uff1a\u76f4\u89d2\u5750\u6807\u65b9\u7a0b<\/a><\/li><li class='ez-toc-page-1 ez-toc-heading-level-3'><a class=\"ez-toc-link ez-toc-heading-17\" href=\"https:\/\/blog.iyatt.com\/?p=21413\/#%E6%96%B9%E6%B3%95%E4%BA%8C%EF%BC%9A%E5%8F%82%E6%95%B0%E6%96%B9%E7%A8%8B-4\" >\u65b9\u6cd5\u4e8c\uff1a\u53c2\u6570\u65b9\u7a0b<\/a><\/li><\/ul><\/li><\/ul><\/li><\/ul><\/nav><\/div>\n<h1><span class=\"ez-toc-section\" id=\"%E5%9B%BE%E5%83%8F\"><\/span>\u56fe\u50cf<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2023\/10\/image-1696770130316.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 572px; --smush-placeholder-aspect-ratio: 572\/400;\" \/><\/p>\n<p>\u4f7f\u7528\u4e0b\u9762\u7684 Python \u4ee3\u7801\u7ed8\u5236\uff0ca \u53d6\u503c 1 \u65f6<br \/>\n(\u672c\u6587\u4ee3\u7801\u90fd\u662f\u5728 Jupyter \u73af\u5883\u4e0b\u8fd0\u884c\u6d4b\u8bd5\u7684)<\/p>\n<pre><code class=\"language-python\">import matplotlib.pyplot as plt\nimport numpy as np\n\nalpha = 1\ntheta = np.linspace(0, 2 * np.pi, 100)\nx = alpha * np.cos(theta)**3\ny = alpha * np.sin(theta)**3\n\nplt.axis(&#039;equal&#039;)\nplt.plot(x, y, color=&#039;r&#039;)\nplt.axhline(y=0, color=&#039;k&#039;)\nplt.axvline(x=0, color=&#039;k&#039;)<\/code><\/pre>\n<h1><span class=\"ez-toc-section\" id=\"%E6%96%B9%E7%A8%8B\"><\/span>\u65b9\u7a0b<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<pre><code class=\"language-katex\">\\begin{array}{l}\n\u6781\u5750\u6807\u65b9\u7a0b\uff1a\\rho=\\frac{1}{(\\cos^\\frac{2}{3}\\theta+\\sin^\\frac{2}{3})^\\frac{3}{2}}\uff08a&gt;0\uff0c\\theta\\in[0,2\\pi]\uff09\\\\\n\\\\\n\u76f4\u89d2\u5750\u6807\u7cfb\u65b9\u7a0b\uff1ax^\\frac{2}{3}+y^\\frac{2}{3}=a^\\frac{2}{3}\uff08a\\gt0\uff09\uff0c\u53ef\u8868\u793a\u4e3a\\ y=(a^\\frac{2}{3} - x^\\frac{2}{3})^\\frac{3}{2}\\\\\n\\\\\n\u53c2\u6570\u65b9\u7a0b\uff1a\\begin{cases}\nx = a\\cos^3\\theta \\\\\ny = a\\sin^3\\theta\n\\end{cases}\n\uff08a&gt;0\uff0c\\theta\\in[0,2\\pi]\uff09\n\\end{array}<\/code><\/pre>\n<h1><span class=\"ez-toc-section\" id=\"%E5%B9%B3%E9%9D%A2%E5%91%A8%E9%95%BF%E8%AE%A1%E7%AE%97\"><\/span>\u5e73\u9762\u5468\u957f\u8ba1\u7b97<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<h2><span class=\"ez-toc-section\" id=\"%E6%96%B9%E6%B3%95%E4%B8%80%EF%BC%9A%E7%9B%B4%E8%A7%92%E5%9D%90%E6%A0%87%E6%96%B9%E7%A8%8B\"><\/span>\u65b9\u6cd5\u4e00\uff1a\u76f4\u89d2\u5750\u6807\u65b9\u7a0b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<ul>\n<li>\u5f27\u5fae\u5206\u53c2\u8003\uff1a<a href=\"https:\/\/blog.iyatt.com\/?p=10934\">https:\/\/blog.iyatt.com\/?p=10934<\/a><\/li>\n<\/ul>\n<pre><code class=\"language-katex\">\\begin{array}{l}\ny&#039;=\\frac{3}{2}(a^\\frac{2}{3} - x^\\frac{2}{3})^\\frac{1}{2}(-\\frac{2}{3}x^{-\\frac{1}{3}})=-(a^\\frac{2}{3} - x^\\frac{2}{3})^\\frac{1}{2}x^{-\\frac{1}{3}} \\\\ \\\\\n\\begin{aligned}\nL&amp;=4\\int_0^a\\sqrt{1+y&#039;^2}dx \\\\\n&amp;=4\\int_0^a\\sqrt{1+(a^\\frac{2}{3} - x^\\frac{2}{3})x^{-\\frac{2}{3}}}dx \\\\\n&amp;=4\\int_0^a|a^\\frac{1}{3}x^{-\\frac{1}{3}}|dx \\\\\n&amp;=4\\times a^\\frac{1}{3}\\times\\frac{3}{2}x^\\frac{2}{3}|_0^a \\\\\n&amp;=6a\n\\end{aligned}\n\\end{array}<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"%E6%96%B9%E6%B3%95%E4%BA%8C%EF%BC%9A%E5%8F%82%E6%95%B0%E6%96%B9%E7%A8%8B\"><\/span>\u65b9\u6cd5\u4e8c\uff1a\u53c2\u6570\u65b9\u7a0b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<pre><code class=\"language-katex\">\\begin{aligned}\nL&amp;=4\\times\\int_0^{\\frac{\\pi}2}\\sqrt{x&#039;^2(\\theta)+y&#039;^2(\\theta))}d\\theta\\\\\n&amp;=4\\times\\int_0^{\\frac{\\pi}2}\\sqrt{[3a\\cos^2\\theta(-\\sin \\theta)]^2+(3a\\sin^2\\theta\\cos \\theta)^2}\\ d\\theta\\\\\n&amp;=4\\times\\int_0^{\\frac{\\pi}2}\\sqrt{9a^2\\sin^2\\theta\\cos^2\\theta(\\sin^2\\theta+\\cos^2\\theta)}d\\theta\\\\\n&amp;=4\\times\\int_0^{\\frac{\\pi}2}3a\\sin\\theta\\cos\\theta d\\theta\\\\\n&amp;=12a\\times\\int_0^{\\frac{\\pi}2}\\sin\\theta d\\sin\\theta\\\\\n&amp;=6a\\times sin^2\\theta|_0^{\\frac{\\pi}2}\\\\\\\\\n&amp;=6a\n\n\\end{aligned}<\/code><\/pre>\n<h1><span class=\"ez-toc-section\" id=\"%E5%B9%B3%E9%9D%A2%E9%9D%A2%E7%A7%AF%E8%AE%A1%E7%AE%97\"><\/span>\u5e73\u9762\u9762\u79ef\u8ba1\u7b97<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<ul>\n<li>\u8ba1\u7b97\u4e2d\u7528\u5230\u7684 Wallis \u516c\u5f0f\u53c2\u8003\uff1a<a href=\"https:\/\/blog.iyatt.com\/?p=11779\">https:\/\/blog.iyatt.com\/?p=11779<\/a><\/li>\n<\/ul>\n<h2><span class=\"ez-toc-section\" id=\"%E6%96%B9%E6%B3%95%E4%B8%80%EF%BC%9A%E7%9B%B4%E8%A7%92%E5%9D%90%E6%A0%87%E6%96%B9%E7%A8%8B-2\"><\/span>\u65b9\u6cd5\u4e00\uff1a\u76f4\u89d2\u5750\u6807\u65b9\u7a0b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<pre><code class=\"language-katex\">\\begin{aligned}\nA&amp;=4\\int_0^aydx \\\\\n&amp;=4\\int_0^a(a^\\frac{2}{3}-x^\\frac{2}{3})^\\frac{3}{2} \\\\\n&amp;\\xlongequal{x=a\\sin^3\\theta}12a\\int_0^\\frac{\\pi}{2}(a^\\frac{2}{3}-a^\\frac{2}{3}\\sin^2\\theta)^\\frac{3}{2}\\sin^2\\theta\\cos\\theta d\\theta \\\\\n&amp;=12a^2\\int_0^\\frac{\\pi}{2}\\cos^4\\theta\\sin^2\\theta d\\theta \\\\\n&amp;=12a^2\\int_0^\\frac{\\pi}{2}\\cos^4\\theta(1-\\cos^2\\theta)d\\theta \\\\\n&amp;=12a^2(\\int_0^\\frac{\\pi}{2}\\cos^4\\theta d\\theta-\\int_0^\\frac{\\pi}{2}\\cos^6\\theta d\\theta) \\\\\n&amp;=12a^2(\\frac{3}{4}\\times\\frac{1}{2}\\times\\frac{\\pi}{2}-\\frac{5}{6}\\times\\frac{3}{4}\\times\\frac{1}{2}\\times\\frac{\\pi}{2}) \\\\\n&amp;=\\frac{3}{8}\\pi a^2\n\\end{aligned}<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"%E6%96%B9%E6%B3%95%E4%BA%8C%EF%BC%9A%E5%8F%82%E6%95%B0%E6%96%B9%E7%A8%8B-2\"><\/span>\u65b9\u6cd5\u4e8c\uff1a\u53c2\u6570\u65b9\u7a0b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<pre><code class=\"language-katex\">\\begin{aligned}\nA&amp;=4\\int_0^aydx \\\\\n&amp;=4\\int_{\\frac{\\pi}2}^0a\\sin^3\\theta\\cdot3a\\cos^2\\theta(-\\sin\\theta)d\\theta\\\\\n&amp;=12a^2\\int_0^{\\frac{\\pi}2}\\sin^4\\theta\\cos^2\\theta d\\theta\\\\\n&amp;=12a^2\\int_0^{\\frac{\\pi}2}\\sin^4\\theta(1-\\sin^2\\theta)d\\theta\\\\\n&amp;=12a^2\\int_0^{\\frac{\\pi}2}(\\sin^4\\theta-\\sin^6\\theta)d\\theta\\\\\n&amp;=12a^2(\\frac{3}{4}\\times\\frac12\\times\\frac{\\pi}2-\\frac56\\times\\frac{3}{4}\\times\\frac12\\times\\frac{\\pi}2)\\\\\n&amp;=\\frac{3\\pi a^2}8\n\\end{aligned}<\/code><\/pre>\n<h1><span class=\"ez-toc-section\" id=\"%E6%97%8B%E8%BD%AC%E4%BD%93\"><\/span>\u65cb\u8f6c\u4f53<span class=\"ez-toc-section-end\"><\/span><\/h1>\n<h2><span class=\"ez-toc-section\" id=\"%E6%97%8B%E8%BD%AC%E5%8A%A8%E7%94%BB\"><\/span>\u65cb\u8f6c\u52a8\u753b<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2023\/10\/EV2023.10.0921.3807.gif\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 576px; --smush-placeholder-aspect-ratio: 576\/526;\" \/><\/p>\n<p>\u65cb\u8f6c\u52a8\u753b\u7531\u4e0b\u9762 Python \u4ee3\u7801\u751f\u6210<\/p>\n<pre><code class=\"language-python\">%matplotlib widget\nimport matplotlib.pyplot as plt\nfrom mpl_toolkits.mplot3d import Axes3D\nimport numpy as np\nfrom matplotlib.animation import FuncAnimation\n\nalpha = 2\n\n# \u661f\u5f62\u7ebf\u53c2\u6570\u65b9\u7a0b\ndef star(t):\n    x = alpha * np.cos(t)**3\n    y = alpha * np.sin(t)**3\n    return x, y\n\n# \u521b\u5efa\u4e00\u4e2a\u4e09\u7ef4\u5750\u6807\u8f74\nfig = plt.figure()\nax = fig.add_subplot(111, projection=&#039;3d&#039;)\n\nt = np.linspace(0, 2 * np.pi, 100)\nx, y = star(t)\n\ncmap = plt.get_cmap(&#039;rainbow&#039;)\n\n# \u5b9a\u4e49\u521d\u59cb\u5316\u51fd\u6570\uff0c\u6e05\u7a7a\u5f53\u524d\u5e27\ndef init():\n    ax.clear()\n    return ax,\n\n# \u5b9a\u4e49\u66f4\u65b0\u51fd\u6570\uff0c\u7ed8\u5236\u6bcf\u4e00\u5e27\u7684\u56fe\u5f62\ndef update(frame):\n    # \u6e05\u7a7a\u5f53\u524d\u5e27\n    ax.clear()\n    # \u8bbe\u7f6e\u5750\u6807\u8f74\u7684\u6807\u7b7e\u548c\u8303\u56f4\n    ax.set_xlabel(&#039;X&#039;)\n    ax.set_ylabel(&#039;Y&#039;)\n    ax.set_zlabel(&#039;Z&#039;)\n    ax.set_xlim(-1.5, 1.5)\n    ax.set_ylim(-1.5, 1.5)\n    ax.set_zlim(-1.5, 1.5)\n    # \u7ed8\u5236\u4e09\u4e2a\u5750\u6807\u8f74\n    ax.plot([0, 5], [0, 0], [0, 0], color=&#039;r&#039;, label=&#039;X&#039;)\n    ax.plot([0, 0], [0, 5], [0, 0], color=&#039;g&#039;, label=&#039;Y&#039;)\n    ax.plot([0, 0], [0, 0], [0, 5], color=&#039;b&#039;, label=&#039;Z&#039;)\n    # \u8ba1\u7b97\u65cb\u8f6c\u540e\u7684y\u548cz\u5750\u6807\n    y_rot = y * np.cos(frame)\n    z_rot = y * np.sin(frame)\n    # \u7ed8\u5236\u65cb\u8f6c\u540e\u7684\u66f2\u7ebf\n    ax.plot(x, y_rot, zs=z_rot, zdir=&#039;z&#039;, color=cmap(frame \/ (2*np.pi)))\n\n# \u521b\u5efa\u52a8\u753b\u5bf9\u8c61\uff0c\u8bbe\u7f6e\u5e27\u6570\uff0c\u95f4\u9694\u548c\u91cd\u590d\u64ad\u653e\nani = FuncAnimation(fig, update, frames=np.linspace(0, 2 * np.pi, 100), init_func=init, interval=10, repeat=True)<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"%E6%97%8B%E8%BD%AC%E4%BD%93%E5%9B%BE%E5%83%8F\"><\/span>\u65cb\u8f6c\u4f53\u56fe\u50cf<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p><img decoding=\"async\" data-src=\"https:\/\/blog.iyatt.com\/wp-content\/uploads\/2023\/10\/image-1696859073309.png\" alt=\"file\" src=\"data:image\/svg+xml;base64,PHN2ZyB3aWR0aD0iMSIgaGVpZ2h0PSIxIiB4bWxucz0iaHR0cDovL3d3dy53My5vcmcvMjAwMC9zdmciPjwvc3ZnPg==\" class=\"lazyload\" style=\"--smush-placeholder-width: 646px; --smush-placeholder-aspect-ratio: 646\/590;\" \/><\/p>\n<p>\u65cb\u8f6c\u4f53\u56fe\u50cf\u7531\u4e0b\u9762 Python \u4ee3\u7801\u751f\u6210<\/p>\n<pre><code class=\"language-python\">%matplotlib widget\nimport matplotlib.pyplot as plt\nfrom mpl_toolkits.mplot3d import Axes3D\nimport numpy as np\n\nalpha = 2\n\n# \u661f\u5f62\u7ebf\u53c2\u6570\u65b9\u7a0b\ndef star(t):\n    x = alpha * np.cos(t)**3\n    y = alpha * np.sin(t)**3\n    return x, y\n\n# \u521b\u5efa\u4e00\u4e2a\u4e09\u7ef4\u5750\u6807\u8f74\nfig = plt.figure()\nax = fig.add_subplot(111, projection=&#039;3d&#039;)\n\nt = np.linspace(0, 2 * np.pi, 100)\nx, y = star(t)\n\ncmap = plt.get_cmap(&#039;rainbow&#039;)\n\n# \u5c06\u661f\u5f62\u7ebf\u4ee5 x \u8f74\u4e3a\u4e2d\u5fc3\u7ebf\u65cb\u8f6c\u4e00\u5468\uff0c\u5f97\u5230\u4e09\u7ef4\u56fe\u5f62\nfor angle in np.linspace(0, 2 * np.pi, 100):\n    # \u8ba1\u7b97\u65cb\u8f6c\u540e\u7684 y \u548c z \u5750\u6807\n    y_rot = y * np.cos(angle)\n    z_rot = y * np.sin(angle)\n    ax.plot(x, y_rot, zs=z_rot, zdir=&#039;z&#039;, color=cmap(angle \/ (2*np.pi)))\n\n# \u8bbe\u7f6e\u5750\u6807\u8f74\u7684\u6807\u7b7e\u548c\u8303\u56f4\nax.set_xlabel(&#039;X&#039;)\nax.set_ylabel(&#039;Y&#039;)\nax.set_zlabel(&#039;Z&#039;)\nax.set_xlim(-1.5, 1.5)\nax.set_ylim(-1.5, 1.5)\nax.set_zlim(-1.5, 1.5)\n\nax.plot([0, 5], [0, 0], [0, 0], color=&#039;r&#039;, label=&#039;X&#039;)\nax.plot([0, 0], [0, 5], [0, 0], color=&#039;g&#039;, label=&#039;Y&#039;)\nax.plot([0, 0], [0, 0], [0, 5], color=&#039;b&#039;, label=&#039;Z&#039;)<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"%E6%97%8B%E8%BD%AC%E4%BD%93%E4%BD%93%E7%A7%AF%E8%AE%A1%E7%AE%97\"><\/span>\u65cb\u8f6c\u4f53\u4f53\u79ef\u8ba1\u7b97<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u661f\u5f62\u7ebf\u65cb\u8f6c\u4f53\u4f53\u79ef\u8ba1\u7b97\u53ef\u4ee5\u770b\u4f5c\u662f\u5706\u67f1\u7684\u4f53\u79ef\u8ba1\u7b97\uff0c\u5c31\u662f\u5e95\u9762\u79ef\u4e58\u4ee5\u9ad8\u3002<br \/>\ny \u8f74\u5de6\u53f3\u4e24\u90e8\u5206\u7684\u4f53\u79ef\u76f8\u7b49\uff0c\u90a3\u4e48\u53ef\u4ee5\u53ea\u8ba1\u7b97\u53f3\u4fa7\u7684\u3002<br \/>\n\u5e95\u9762\u5706\u534a\u5f84\u5c31\u662f y\uff0c\u5e95\u9762\u79ef\u5c31\u662f <code class=\"katex-inline\">\\pi y^2<\/code>\uff0c\u9ad8\u662f\u5bf9\u5f27\u5fae\u5206\u8fdb\u884c 0 \u5230 a \u7684\u79ef\u5206\uff08y \u8f74\u53f3\u534a\u90e8\u5206\uff09\uff0c\u518d\u4e58\u4ee5 2 \u5c31\u662f\u8981\u8ba1\u7b97\u7684\u4f53\u79ef\u3002<\/p>\n<h3><span class=\"ez-toc-section\" id=\"%E6%96%B9%E6%B3%95%E4%B8%80%EF%BC%9A%E7%9B%B4%E8%A7%92%E5%9D%90%E6%A0%87%E6%96%B9%E7%A8%8B-3\"><\/span>\u65b9\u6cd5\u4e00\uff1a\u76f4\u89d2\u5750\u6807\u65b9\u7a0b<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<pre><code class=\"language-katex\">\\begin{array}{l}\ny^2=(a^\\frac{2}{3} - x^\\frac{2}{3})^3 \\\\\n\\begin{aligned}\nV&amp;=2\\pi\\int_0^ay^2dx \\\\\n&amp;=2\\pi\\int_0^a(a^\\frac{2}{3} - x^\\frac{2}{3})^3dx \\\\\n&amp;\\xlongequal{x=a\\sin^3\\theta}2\\pi\\int_0^a(a^\\frac{2}{3}-a^\\frac{2}{3}\\sin^2\\theta)^3\\cdot3a\\sin^2\\theta\\cos\\theta d\\theta \\\\\n&amp;=6\\pi a^3\\int_0^a\\cos^7\\theta\\sin^2\\theta d\\theta \\\\\n&amp;=6\\pi a^3\\int_0^a\\cos^7\\theta(1-\\cos^2\\theta)d\\theta \\\\\n&amp;=6\\pi a^3\\int_0^a(\\cos^7\\theta-\\cos^9\\theta)d\\theta \\\\\n&amp;\\xlongequal{Wallis\u516c\u5f0f}6\\pi a^3(\\frac{6\\times4\\times2}{7\\times5\\times3}-\\frac{8\\times6\\times4\\times2}{9\\times7\\times5\\times3}) \\\\\n&amp;=\\frac{32}{105}\\pi a^3\n\\end{aligned}\n\\end{array}<\/code><\/pre>\n<h3><span class=\"ez-toc-section\" id=\"%E6%96%B9%E6%B3%95%E4%BA%8C%EF%BC%9A%E5%8F%82%E6%95%B0%E6%96%B9%E7%A8%8B-3\"><\/span>\u65b9\u6cd5\u4e8c\uff1a\u53c2\u6570\u65b9\u7a0b<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<pre><code class=\"language-katex\">\\begin{aligned}\nV&amp;=2\\pi\\int_0^ay^2dx\\\\\n&amp;=2\\pi\\int_{\\frac{\\pi}2}^0a^2\\sin^6\\theta\\cdot3a\\cos^2\\theta(-\\sin\\theta)d\\theta\\\\\n&amp;=6\\pi a^3\\int_0^{\\frac{\\pi}2}\\sin^7\\theta\\cos^2\\theta d\\theta\\\\\n&amp;=6\\pi a^3\\int_0^{\\frac{\\pi}2}\\sin^7\\theta(1-\\sin^2\\theta)d\\theta\\\\\n&amp;=6\\pi a^3\\int_0^{\\frac{\\pi}2}(\\sin^7\\theta-\\sin^9\\theta)d\\theta\\\\\n&amp;=6\\pi a^3(\\frac67\\times\\frac45\\times\\frac23-\\frac89\\times\\frac67\\times\\frac45\\times\\frac23)\\\\\n&amp;=\\frac{32}{105}\\pi a^3\n\\end{aligned}<\/code><\/pre>\n<h2><span class=\"ez-toc-section\" id=\"%E6%97%8B%E8%BD%AC%E4%BD%93%E4%BE%A7%E9%9D%A2%E7%A7%AF%E8%AE%A1%E7%AE%97\"><\/span>\u65cb\u8f6c\u4f53\u4fa7\u9762\u79ef\u8ba1\u7b97<span class=\"ez-toc-section-end\"><\/span><\/h2>\n<p>\u661f\u5f62\u7ebf\u65cb\u8f6c\u4f53\u4fa7\u9762\u79ef\u8ba1\u7b97\u53ef\u4ee5\u770b\u4f5c\u662f\u5706\u67f1\u7684\u4fa7\u9762\u79ef\u8ba1\u7b97\uff0c\u5c31\u662f\u5e95\u9762\u5706\u5468\u957f\u4e58\u4ee5\u9ad8\u3002<br \/>\n\u5e95\u9762\u5706\u534a\u5f84\u5c31\u662f y\uff0c\u5e95\u9762\u5468\u957f\u5c31\u662f <code class=\"katex-inline\">2\\pi y<\/code>\uff0c\u9ad8\u662f\u5bf9\u5f27\u5fae\u5206\u8fdb\u884c 0 \u5230 a \u7684\u79ef\u5206\uff08y \u8f74\u53f3\u534a\u90e8\u5206\uff09\uff0c\u79ef\u5206\u7ed3\u679c\u5c31\u662f\u53f3\u534a\u90e8\u5206\uff0c\u518d\u4e58\u4ee5 2 \u5c31\u662f\u8981\u8ba1\u7b97\u7684\u4fa7\u9762\u79ef\u3002<\/p>\n<h3><span class=\"ez-toc-section\" id=\"%E6%96%B9%E6%B3%95%E4%B8%80%EF%BC%9A%E7%9B%B4%E8%A7%92%E5%9D%90%E6%A0%87%E6%96%B9%E7%A8%8B-4\"><\/span>\u65b9\u6cd5\u4e00\uff1a\u76f4\u89d2\u5750\u6807\u65b9\u7a0b<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<pre><code class=\"language-katex\">\\begin{aligned}\nS&amp;=2\\times2\\pi\\int_0^ay\\sqrt{1+y&#039;^2}dx \\\\\n&amp;=4\\pi\\int_0^a|(a^\\frac{2}{3} - x^\\frac{2}{3})^\\frac{3}{2}|\\sqrt{1+(a^\\frac{2}{3} - x^\\frac{2}{3})x^{-\\frac{2}{3}}}dx \\\\\n&amp;\\xlongequal{x=a\\sin^3\\theta}4\\pi\\int_0^\\frac{\\pi}{2}|a\\cos^3\\theta|\\sqrt{1+\\cos^2\\theta\\sin^{-2}\\theta}\\times3a\\sin^2\\theta\\cos\\theta d\\theta \\\\\n&amp;=12\\pi a^2\\int_0^\\frac{\\pi}{2}\\cos^4\\theta\\sin^2\\theta\\sqrt{1+\\cot^2\\theta}d\\theta \\\\\n&amp;=12\\pi a^2\\int_0^\\frac{\\pi}{2}\\cos^4\\theta\\sin^2\\theta|\\csc\\theta|d\\theta \\\\\n&amp;=12\\pi a^2\\int_0^\\frac{\\pi}{2}\\cos^4\\theta|\\sin\\theta|d\\theta \\\\\n&amp;=-12\\pi a^2\\int_0^\\frac{\\pi}{2}\\cos^4\\theta d\\cos\\theta \\\\\n&amp;=-12\\pi a^2\\cdot\\frac{1}{5}\\cos^5\\theta|_0^\\frac{\\pi}{2} \\\\\n&amp;=\\frac{12}{5}\\pi a^2\n\\end{aligned}<\/code><\/pre>\n<h3><span class=\"ez-toc-section\" id=\"%E6%96%B9%E6%B3%95%E4%BA%8C%EF%BC%9A%E5%8F%82%E6%95%B0%E6%96%B9%E7%A8%8B-4\"><\/span>\u65b9\u6cd5\u4e8c\uff1a\u53c2\u6570\u65b9\u7a0b<span class=\"ez-toc-section-end\"><\/span><\/h3>\n<pre><code class=\"language-katex\">\\begin{aligned}\nS&amp;=2\\times2\\pi\\int_0^{\\frac\\pi2}y(t)\\sqrt{[x&#039;^2(t)]+[y&#039;^2(t)]}\\ dt\\\\\n&amp;=4\\pi\\int_0^\\frac\\pi2a\\sin^3t\\sqrt{[3a\\cos^2t(-\\sin t)]^2+(3a\\sin^2t\\cos t)^2}dt\\\\\n&amp;=4\\pi\\int_0^\\frac\\pi2a\\sin^3t\\sqrt{9a^2\\sin^2t\\cos^2t(\\sin^2t+\\cos^2t)}\\\\\n&amp;=12\\pi a^2\\int_0^\\frac\\pi2\\sin^4t\\cos tdt\\\\\n&amp;=12\\pi a^2\\int_0^\\frac\\pi2\\sin^4td\\sin t\\\\\n&amp;=12\\pi a^2\\times\\frac15\\sin^5t|_0^\\frac\\pi2\\\\\n&amp;=\\frac{12}5\\pi a^2\n\\end{aligned}<\/code><\/pre>\n","protected":false},"excerpt":{"rendered":"<p>\u56fe\u50cf \u4f7f\u7528\u4e0b\u9762\u7684 Python \u4ee3\u7801\u7ed8\u5236\uff0ca \u53d6\u503c 1 \u65f6 (\u672c\u6587\u4ee3\u7801\u90fd\u662f\u5728 Jupyter \u73af\u5883\u4e0b\u8fd0\u884c\u6d4b\u8bd5\u7684 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"zakra_page_container_layout":"customizer","zakra_page_sidebar_layout":"customizer","zakra_remove_content_margin":false,"zakra_sidebar":"customizer","zakra_transparent_header":"customizer","zakra_logo":0,"zakra_main_header_style":"default","zakra_menu_item_color":"","zakra_menu_item_hover_color":"","zakra_menu_item_active_color":"","zakra_menu_active_style":"","zakra_page_header":true,"_lmt_disableupdate":"no","_lmt_disable":"no","footnotes":""},"categories":[1,606,592,612],"tags":[949,909,1071,1068,1070,1073,950,1069,1074,1076,1077,1075,1067,1072],"class_list":["post-21413","post","type-post","status-publish","format-standard","hentry","category-all","category-jupyter","category-python","category-612","tag-jupyter","tag-python","tag-1071","tag-1068","tag-1070","tag-1073","tag-950","tag-1069","tag-1074","tag-1076","tag-1077","tag-1075","tag-1067","tag-1072"],"modified_by":"IYATT-yx","_links":{"self":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/21413","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=21413"}],"version-history":[{"count":19,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/21413\/revisions"}],"predecessor-version":[{"id":21452,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=\/wp\/v2\/posts\/21413\/revisions\/21452"}],"wp:attachment":[{"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=21413"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=21413"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/blog.iyatt.com\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=21413"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}