{"id":6429,"date":"2023-06-01T16:58:42","date_gmt":"2023-06-01T07:58:42","guid":{"rendered":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/?page_id=6429"},"modified":"2024-06-13T10:38:00","modified_gmt":"2024-06-13T01:38:00","slug":"%e7%a9%8d%e5%88%86%ef%bc%9a%e3%81%84%e3%81%8f%e3%81%a4%e3%81%8b%e3%81%ae%e5%bf%9c%e7%94%a8%e3%81%ae%e4%be%8b%e9%a1%8c","status":"publish","type":"page","link":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/%e7%a9%8d%e5%88%86%ef%bc%9a%e3%81%84%e3%81%8f%e3%81%a4%e3%81%8b%e3%81%ae%e5%bf%9c%e7%94%a8\/%e7%a9%8d%e5%88%86%ef%bc%9a%e3%81%84%e3%81%8f%e3%81%a4%e3%81%8b%e3%81%ae%e5%bf%9c%e7%94%a8%e3%81%ae%e4%be%8b%e9%a1%8c\/","title":{"rendered":"\u7a4d\u5206\uff1a\u3044\u304f\u3064\u304b\u306e\u5fdc\u7528\u306e\u4f8b\u984c"},"content":{"rendered":"<p><!--more--><\/p>\n<h3>\u4f8b\u984c 1.<\/h3>\n<p>\u66f2\u7dda\uff08\u5b9f\u969b\u306b\u306f\u76f4\u7dda\u3060\u304c\uff09$y = x$ \u306e $0 \\le x \\le 1$ \u306e\u9577\u3055\u3092\u6c42\u3081\u3088\u3002<\/p>\n<p>\u30d2\u30f3\u30c8\uff1a\u7a4d\u5206\u3092\u3057\u306a\u304f\u3066\u3082 $(0,0)$ \u3068 $(1, 1)$ \u3092\u3080\u3059\u3076\u76f4\u7dda\u306e\u9577\u3055\u3060\u304b\u3089\u7b54\u3048\u306f $\\sqrt{1^2 + 1^2} = \\sqrt{2}$\u3002<\/p>\n<h3>\u4f8b\u984c 2.<\/h3>\n<p>\u66f2\u7dda\uff08\u3053\u308c\u306f\u653e\u7269\u7dda\uff09$y = x^2$ \u306e $0 \\le x \\le 1$ \u306e\u9577\u3055\u3092\u6c42\u3081\u3088\u3002<\/p>\n<p>\u30d2\u30f3\u30c8\uff1a\u7121\u7406\u95a2\u6570\u306e\u7a4d\u5206\u3067 $\\displaystyle \\int \\sqrt{1+x^2}\\, dx $ \u306f<span style=\"font-family: helvetica, arial, sans-serif;\"><strong><a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/%e7%90%86%e5%b7%a5%e7%b3%bb%e3%81%ae%e6%95%b0%e5%ad%a6b\/%e7%84%a1%e7%90%86%e9%96%a2%e6%95%b0%e3%81%ae%e7%a9%8d%e5%88%86\/#_3_displaystyle_int_sqrtx21_dx\" target=\"_blank\" rel=\"noopener\">\u308f\u304b\u3063\u3066\u3044\u308b<\/a><\/strong><\/span>\u3002\u3058\u3083\u3042\uff0c$\\displaystyle \\int \\sqrt{1+(2 x)^2}\\, dx $ \u3082\u3067\u304d\u308b\u3088\u306d\u3002<\/p>\n<h3>\u4f8b\u984c 3.<\/h3>\n<p>\u30ab\u30c6\u30ca\u30ea\u30fc\u66f2\u7dda $\\displaystyle\u00a0 y = \\frac{1}{a} \\left( \\cosh( ax ) -1 \\right) + h$ \u306e $0 \\le x \\le \\frac{1}{a}$ \u306e\u9577\u3055\u3092\u6c42\u3081\u3088\u3002<\/p>\n<p>\\begin{eqnarray}<br \/>\n\\frac{dy}{dx} &amp;=&amp; \\sinh (ax) \\\\<br \/>\n\\therefore\\ \\ L &amp;=&amp; \\int_0^{\\frac{1}{a}} \\sqrt{1 + \\left(\\frac{dy}{dx}\\right)^2}\\, dx \\\\<br \/>\n&amp;=&amp; \\cdots<br \/>\n\\end{eqnarray}<\/p>\n<p>\u53cc\u66f2\u7dda\u95a2\u6570\u306e\u95a2\u4fc2 $\\cosh^2 t &#8211; \\sinh^2 t = 1$ \u3082\u5fd8\u308c\u306a\u3044\u3067&#8230;<\/p>\n<ul>\n<li>\u53c2\u8003\uff1a<a href=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/3189\/\" target=\"_blank\" rel=\"noopener\">\u30ab\u30c6\u30ca\u30ea\u30fc\u66f2\u7dda<\/a><\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-6603\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_6965-640x308.png\" alt=\"\" width=\"640\" height=\"308\" srcset=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_6965-640x308.png 640w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_6965-300x144.png 300w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_6965-1536x739.png 1536w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_6965-2048x986.png 2048w, https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/IMG_6965-750x361.png 750w\" sizes=\"auto, (max-width: 640px) 100vw, 640px\" \/><\/p>\n<h3>\u4f8b\u984c 4.<\/h3>\n<p>\u9577\u534a\u5f84 $a$, \u77ed\u534a\u5f84 $b$ \u306e\u6955\u5186\u306e\u9762\u7a4d\u3092\u6c42\u3081\u3088\u3002<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"aligncenter size-large wp-image-8899\" src=\"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-content\/uploads\/sites\/76\/SPBDaen01.svg\" alt=\"\" width=\"640\" height=\"383\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>$\\displaystyle \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ \u306e\u6955\u5186\u306e\u9762\u7a4d\u3002<\/p>\n<p>$a &gt; b &gt; 0$ \u3068\u3057\u3066 $a$ \u306f\u9577\u534a\u5f84\uff0c$b$ \u306f\u77ed\u534a\u5f84\u3068\u547c\u3076\u3002\u96e2\u5fc3\u7387 $e$ \u3092\u4f7f\u3063\u3066\u66f8\u304f\u3068\uff0c$b = a \\sqrt{1-e^2}$\u3002<\/p>\n<p>$\\displaystyle \\frac{x^2}{a^2} + \\frac{y^2}{b^2} = 1$ \u3088\u308a $\\displaystyle y = \\pm b \\sqrt{1-\\frac{x^2}{a^2}}$\u3002\u4e0a\u534a\u5206\u306e\u9762\u7a4d\uff0c\u3064\u307e\u308a $\\displaystyle y = b \\sqrt{1-\\frac{x^2}{a^2}}$ \u3068 $x$ \u8ef8\u3067\u56f2\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u3092\u6c42\u3081\u30662\u500d\u3059\u308c\u3070\u3088\u3044\u304b\u3089\uff0c<\/p>\n<p>\\begin{eqnarray}<br \/>\nS &amp;=&amp; 2 \\int_{-a}^a b \\sqrt{1 \u2013\\frac{x^2}{a^2}} dx \\\\<br \/>\n&amp;&amp;\\qquad (x \\equiv a \\sin\\theta, \\ \\sqrt{1 -\\frac{x^2}{a^2}} = \\cos\\theta, \\ dx = a \\cos\\theta d\\theta)\\\\<br \/>\n&amp;=&amp; 2 a b \\int_{-\\pi\/2}^{\\pi\/2} \\cos^2\\theta d\\theta\\\\<br \/>\n&amp;=&amp; a b \\int_{-\\pi\/2}^{\\pi\/2} (1 + \\cos 2\\theta) d\\theta \\\\<br \/>\n&amp;=&amp;a b \\left[ \\theta + \\frac{1}{2}\\sin 2\\theta\\right]_{-\\pi\/2}^{\\pi\/2} \\\\<br \/>\n&amp;=&amp; \\pi a b<br \/>\n\\end{eqnarray}<\/p>\n","protected":false},"excerpt":{"rendered":"","protected":false},"author":33,"featured_media":0,"parent":2158,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"inline_featured_image":false,"footnotes":""},"class_list":["post-6429","page","type-page","status-publish","hentry","nodate","item-wrap"],"aioseo_notices":[],"_links":{"self":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6429","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/users\/33"}],"replies":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/comments?post=6429"}],"version-history":[{"count":11,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6429\/revisions"}],"predecessor-version":[{"id":8900,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/6429\/revisions\/8900"}],"up":[{"embeddable":true,"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/pages\/2158"}],"wp:attachment":[{"href":"https:\/\/home.hirosaki-u.ac.jp\/relativity\/wp-json\/wp\/v2\/media?parent=6429"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}