{"id":382,"date":"2012-04-27T18:44:06","date_gmt":"2012-04-27T09:44:06","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=382"},"modified":"2021-09-24T17:53:21","modified_gmt":"2021-09-24T08:53:21","slug":"tok201206","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok201206\/","title":{"rendered":"\u6771\u5de5\u59272012\uff1a\u7b2c6\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(xyz\\) \u7a7a\u9593\u306b \\(4\\) \u70b9 P \\((0,0,2)\\) , A \\((0,2,0)\\) , B \\((\\sqrt{3},-1,0)\\) , C \\((-\\sqrt{3},-1,0)\\) \u3092\u3068\u308b.\r\n\u56db\u9762\u4f53 PABC \u306e \\(x^2+y^2 \\geqq 1\\) \u3092\u307f\u305f\u3059\u90e8\u5206\u306e\u4f53\u7a4d\u3092\u6c42\u3081\u3088.<\/p>\r\n<hr>\r\n<!--more-->\r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p>\u6c42\u3081\u308b\u4f53\u7a4d\u3092 \\(V\\) \u3068\u304a\u304f.<br \/>\r\n\u5bfe\u79f0\u6027\u304b\u3089, \u4e0b\u56f3\u659c\u7dda\u90e8\u306e\u90e8\u5206\u3092 \\(D\\) \u3068\u3057, \u305d\u306e\u4f53\u7a4d\u3092 \\(W\\) \u3068\u304a\u3051\u3070\r\n\\[\r\nV = 6W\r\n\\]\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/toko_2012_06_01.png\" alt=\"\" title=\"toko_2012_06_01\" class=\"aligncenter size-full\" \/>\r\n<p>\u9818\u57df \\(D\\) \u306e\u5e73\u9762 \\(y =-t \\quad \\left( \\dfrac{1}{2} \\leqq t \\leqq 1 \\right)\\) \u3067\u306e\u5207\u308a\u53e3\u306f, \u4e0b\u56f3\u306e\u9577\u65b9\u5f62 QRST \u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/toko_2012_06_02.png\" alt=\"\" title=\"toko_2012_06_02\" class=\"aligncenter size-full\" \/>\r\n<p>\u3053\u306e\u9762\u7a4d\u3092 \\(U(t)\\) \u3068\u304a\u304f. \u5404\u70b9\u306e\u5ea7\u6a19\u306b\u3064\u3044\u3066\r\n\\[\r\n\\text{Q} \\ \\left( \\sqrt{1-t^2} , -t , 0 \\right) , \\quad \\text{R} \\ \\left( \\sqrt{3} t , -t , 0 \\right)\r\n\\]\r\n\u307e\u305f, S \u306f PB \u3092 \\(1 -t : t\\) \u306b\u5185\u5206\u3059\u308b\u306e\u3067\r\n\\[\r\n\\text{S} \\ \\left( \\sqrt{3} (1-t) , -t , 2(1-t) \\right)\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\nU(t) & = \\left( \\sqrt{3} t -\\sqrt{1-t^2} \\right) \\cdot 2(1-t) \\\\\r\n& = 2\\sqrt{3} ( t -t^2 ) -2\\sqrt{1-t^2} +2t \\sqrt{1-t^2}\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nW & = \\displaystyle\\int _ {\\frac{1}{2}}^1 U(t) \\, dt \\\\\r\n& = 2\\sqrt{3} \\displaystyle\\int _ {\\frac{1}{2}}^1 ( t-t^2 ) \\, dt -2 \\underline{\\displaystyle\\int _ {\\frac{1}{2}}^1 \\sqrt{1-t^2} \\, dt} _ {[1]} \\\\\r\n& \\qquad -\\displaystyle\\int _ {\\frac{1}{2}}^1 \\sqrt{1-t^2} \\left( 1-t^2 \\right)' \\, dt\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, [1] \u306f\u4e0b\u56f3\u659c\u7dda\u90e8\u306e\u9762\u7a4d\u3068\u7b49\u3057\u3044\u306e\u3067<\/p>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/toko_2012_06_03.png\" alt=\"\" title=\"toko_2012_06_03\" class=\"aligncenter size-full\" \/>\r\n<p>\\[\\begin{align}\r\n[1] & = \\dfrac{1}{2} \\cdot 1^2 \\cdot \\dfrac{\\pi}{3} -\\dfrac{1}{2} \\cdot \\dfrac{1}{2} \\cdot \\dfrac{\\sqrt{3}}{2} \\\\\r\n& = \\dfrac{\\pi}{6} -\\dfrac{\\sqrt{3}}{8}\r\n\\end{align}\\]\r\n\u3086\u3048\u306b\r\n\\[\\begin{align}\r\nW & = 2\\sqrt{3} \\left[ \\dfrac{t^2}{2} -\\dfrac{t^3}{3} \\right] _ {\\frac{1}{2}}^1 \\\\\r\n& \\qquad -2\\left( \\dfrac{\\pi}{6} -\\dfrac{\\sqrt{3}}{8} \\right) -\\left[ \\dfrac{2 \\left( 1-t^2 \\right)^{\\frac{3}{2}}}{3} \\right] _ {\\frac{1}{2}}^1 \\\\\r\n& = 2\\sqrt{3} \\left( \\dfrac{1}{6} -\\dfrac{1}{12} \\right) -\\dfrac{\\pi}{3} +\\dfrac{\\sqrt{3}}{4} +\\dfrac{2}{3} \\cdot \\dfrac{3\\sqrt{3}}{8} \\\\\r\n& = \\dfrac{2 \\sqrt{3}}{3} -\\dfrac{\\pi}{3}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u4f53\u7a4d\u306f\r\n\\[\r\nV = 6 \\left( \\dfrac{2 \\sqrt{3}}{3} -\\dfrac{\\pi}{3} \\right) = \\underline{4 \\sqrt{3} -2 \\pi}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xyz\\) \u7a7a\u9593\u306b \\(4\\) \u70b9 P \\((0,0,2)\\) , A \\((0,2,0)\\) , B \\((\\sqrt{3},-1,0)\\) , C \\((-\\sqrt{3},-1,0)\\) \u3092\u3068\u308b. \u56db\u9762\u4f53 PA &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/tok201206\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[61],"tags":[141,68],"class_list":["post-382","post","type-post","status-publish","format-standard","hentry","category-toko_2012","tag-toko","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/382","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=382"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/382\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=382"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=382"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=382"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}