{"id":1265,"date":"2015-09-03T23:01:49","date_gmt":"2015-09-03T14:01:49","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1265"},"modified":"2021-09-13T19:15:50","modified_gmt":"2021-09-13T10:15:50","slug":"iks201503","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/iks201503\/","title":{"rendered":"\u533b\u79d1\u6b6f\u79d1\u59272015\uff1a\u7b2c3\u554f"},"content":{"rendered":"
\n

\u5ea7\u6a19\u5e73\u9762\u4e0a\u3067\u6b21\u306e\u3088\u3046\u306b\u5a92\u4ecb\u5909\u6570\u8868\u793a\u3055\u308c\u308b\u66f2\u7dda \\(C\\) \u3092\u8003\u3048\u308b.\r\n\\[\r\n\\left\\{ \\begin{array}{l} x = | \\cos t | \\cos^3 t \\\\ y = | \\sin t | \\sin^3 t \\end{array} \\right. \\quad ( 0 \\leqq t \\leqq 2 \\pi )\r\n\\]\r\n\u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u6b21\u306e\u6761\u4ef6 (\uff0a) \u3092\u6e80\u305f\u3059\u7b2c \\(1\\) \u8c61\u9650\u5185\u306e\u5b9a\u70b9 F \u306e\u5ea7\u6a19\u3092\u6c42\u3081\u3088.<\/p>\r\n

      \r\n
    1. (\uff0a) \u7b2c \\(1\\) \u8c61\u9650\u5185\u3067 \\(C\\) \u4e0a\u306b\u3042\u308b\u3059\u3079\u3066\u306e\u70b9 P \u306b\u3064\u3044\u3066, P \u304b\u3089\u76f4\u7dda \\(x+y = 0\\) \u306b\u4e0b\u308d\u3057\u305f\u5782\u7dda\u3092 PH \u3068\u3059\u308b\u3068\u304d, \u3064\u306d\u306b \\(\\text{PF} = \\text{PH}\\) \u3068\u306a\u308b.<\/li>\r\n<\/ol><\/li>\r\n

    2. (2)<\/strong>\u3000\u70b9 P \u304c \\(C\\) \u5168\u4f53\u3092\u52d5\u304f\u3068\u304d, P \u3068 (1)<\/strong> \u306e\u5b9a\u70b9 F \u3092\u7d50\u3076\u7dda\u5206 PF \u304c\u901a\u904e\u3059\u308b\u9818\u57df\u3092\u56f3\u793a\u3057, \u305d\u306e\u9762\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

    3. (3)<\/strong>\u3000(2)<\/strong> \u306e\u9818\u57df\u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u7acb\u4f53\u306e\u4f53\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

      \r\n\r\n

      \u89e3\u7b54<\/h2>\r\n

      (1)<\/strong><\/p>\r\n

      \\(C\\) \u4e0a\u306e\u70b9 P \u306e \\(x\\) \u5ea7\u6a19, \\(y\\) \u5ea7\u6a19\u306e\u7b26\u53f7\u306f, \u305d\u308c\u305e\u308c \\(\\cos t , \\sin t\\) \u306b\u4e00\u81f4\u3059\u308b\u306e\u3067, \\(0 \\leqq t \\leqq \\dfrac{\\pi}{2}\\) \u306b\u3064\u3044\u3066\u8003\u3048\u308c\u3070\u3088\u3044.
      \r\n\\(x = x(t)\\) , \\(y = y(t)\\) \u3068\u8868\u3059.
      \r\n\u3053\u306e\u3068\u304d, \\(\\cos t \\geqq 0\\) , \\(\\sin t \\geqq 0\\) \u306a\u306e\u3067\r\n\\[\r\nx(t) = \\cos^4 t , \\ y(t) = \\sin^4 t \\ .\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(\\sin^2 t +\\cos^2 t = 1\\) \u306b\u6ce8\u610f\u3059\u308c\u3070, \\(C\\) \u306e\u5f0f\u306f\r\n\\[\r\n\\sqrt{x} +\\sqrt{y} = 1 \\quad ... [1] \\ .\r\n\\]\r\n\"iks20150301\"\r\n

      \\(x , y\\) \u306b\u3064\u3044\u3066\u5bfe\u79f0\u306a\u306e\u3067, \\(C\\) \u306f\u76f4\u7dda \\(\\ell : \\ y = x\\) \u306b\u3064\u3044\u3066\u5bfe\u79f0\u3067\u3042\u308b\u304b\u3089, F \u306f \\(\\ell\\) \u4e0a\u306b\u3042\u308b.\r\n\\[\\begin{align}\r\n\\cos^4 t & = \\sin^4 t \\\\\r\n\\text{\u2234} \\quad \\cos t & = \\sin t \\\\\r\n\\text{\u2234} \\quad t & = \\dfrac{1}{\\sqrt{2}} \\ .\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(\\left( \\dfrac{1}{\\sqrt{2}} \\right)^4 = \\dfrac{1}{4}\\) \u304b\u3089, \\(C\\) \u3068 \\(\\ell\\) \u306e\u4ea4\u70b9\u306f\r\n\\[\r\n\\left( \\dfrac{1}{4} , \\dfrac{1}{4} \\right) \\ .\r\n\\]\r\n\\(\\text{PF} = \\text{PH}\\) \u306a\u306e\u3067\r\n\\[\r\n\\text{F} \\ \\underline{\\left( \\dfrac{1}{2} , \\dfrac{1}{2} \\right)} \\ .\r\n\\]\r\n\u5b9f\u969b\u306b, [1] \u3088\u308a\r\n\\[\\begin{align}\r\nx +y & = 1 -2 \\sqrt{xy} , \\\\\r\nx^2 +y^2 & = \\left( 1 -2 \\sqrt{xy} \\right)^2 -2xy \\\\\r\n& = 2xy -4 \\sqrt{xy} +1 \\ .\r\n\\end{align}\\]\r\n\u3053\u308c\u3089\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n\\text{PH} & = \\dfrac{\\left| 1 \\cdot x +1 \\cdot y \\right|}{\\sqrt{1^2+1^2}} = \\dfrac{x+y}{\\sqrt{2}} \\\\\r\n& = \\dfrac{1 -2\\sqrt{xy}}{\\sqrt{2}} , \\\\\r\n\\text{PF} & = \\sqrt{\\left( x -\\dfrac{1}{2} \\right)^2 +\\left( y -\\dfrac{1}{2} \\right)^2} \\\\\r\n& = \\sqrt{x^2 +y^2 -x -y +\\dfrac{1}{2}} \\\\\r\n& = \\dfrac{\\sqrt{4xy -4 \\sqrt{xy} +1}}{\\sqrt{2}} \\\\\r\n& = \\dfrac{1 -2\\sqrt{xy}}{\\sqrt{2}} \\ .\r\n\\end{align}\\]\r\n\u3059\u306a\u308f\u3061, \\(\\text{PF} = \\text{PH}\\) \u304c\u6210\u7acb\u3057\u3066\u3044\u308b.<\/p>\r\n

      (2)<\/strong><\/p>\r\n

      \\(C\\) \u306e\u3046\u3061, \u7b2c \\(2\\) \u8c61\u9650\u306b\u3042\u308b\u90e8\u5206\u306b\u3064\u3044\u3066, \\(\\dfrac{\\pi}{2} \\leqq t \\leqq \\pi\\) \u306e\u3068\u304d\u3067\u3042\u308a, \\(t = t' +\\dfrac{\\pi}{2} \\ \\left( 0 \\leqq t' \\leqq \\dfrac{\\pi}{2} \\right)\\) \u3068\u304a\u3051\u3070\r\n\\[\\begin{align}\r\nx(t) & = \\sin^4 \\left( t' +\\dfrac{\\pi}{2} \\right) = \\cos^4 t' = y(t') \\\\\r\ny(t) & = -\\cos^4 \\left( t' +\\dfrac{\\pi}{2} \\right) = -\\sin^4 t' = -x(t') \\ .\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(C\\) \u306e\u3046\u3061, \u7b2c \\(1\\) \u8c61\u9650\u306b\u3042\u308b\u90e8\u5206\u3092, \u539f\u70b9\u4e2d\u5fc3\u306b \\(\\dfrac{\\pi}{2}\\) \u56de\u8ee2\u3055\u305b\u305f\u66f2\u7dda\u3068\u306a\u308b.
      \r\n\u7b2c \\(3 , 4\\) \u8c61\u9650\u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306a\u306e\u3067, \\(C\\) \u306e\u6982\u5f62\u306f\u4e0b\u56f3\u306e\u901a\u308a\u3068\u306a\u308b.<\/p>\r\n\"iks20150302\"\r\n

      \u3053\u306e\u3068\u304d, \u7dda\u5206 PF \u304c\u901a\u904e\u3059\u308b\u9818\u57df \\(D\\) \u306f\u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u5883\u754c\u3092\u542b\u3080\uff09\u3068\u306a\u308b.<\/p>\r\n\"iks20150303\"\r\n

      \\(D\\) \u306e\u3046\u3061, \u7b2c \\(1, \\cdots , 4\\) \u8c61\u9650\u306b\u542b\u307e\u308c\u308b\u90e8\u5206\u306e\u9762\u7a4d\u3092\u305d\u308c\u305e\u308c \\(S _ 1 , \\cdots , S _ 4\\) \u3068\u304a\u3051\u3070, \u5bfe\u79f0\u6027\u3092\u5229\u7528\u3057\u3066\r\n\\[\\begin{align}\r\nS _ 1 & = \\dfrac{1}{2} \\cdot 1 \\cdot 1 = \\dfrac{1}{2} , \\\\\r\nS _ 3 & = \\displaystyle\\int _ 0^1 \\left( x -2 \\sqrt{x} +1 \\right) \\, dx \\\\\r\n& = \\left[ \\dfrac{x^2}{2} -\\dfrac{4 x^{\\frac{3}{2}}}{3} +x \\right] _ 0^1 = \\dfrac{1}{6} , \\\\\r\nS _ 2 = S _ 4 & = \\dfrac{S _ 3 +\\left( \\frac{1}{4} \\right)^2}{2} +\\dfrac{1}{2} \\cdot \\dfrac{3}{4} \\cdot \\dfrac{1}{4} \\\\\r\n& = \\dfrac{1}{2} \\left( \\dfrac{11}{48} +\\dfrac{3}{16} \\right) = \\dfrac{1}{2} \\cdot \\dfrac{5}{12} \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u9762\u7a4d \\(S\\) \u306f\r\n\\[\\begin{align}\r\nS & = S _ 1 +S _ 2 +S _ 3 +S _ 4 \\\\\r\n& = \\dfrac{1}{2} +\\dfrac{1}{6} +\\dfrac{5}{12} \\\\\r\n& = \\underline{\\dfrac{13}{12}} \\ .\r\n\\end{align}\\]\r\n

      (3)<\/strong><\/p>\r\n

      \u4e0b\u56f3\u306e\u9818\u57df \\(D _ a , D _ b , D _ c\\) \u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3057\u305f\u7acb\u4f53\u306e\u4f53\u7a4d\u3092 \\(V _ a , V _ b , V _ c\\) \u3068\u304a\u304f\u3068<\/p>\r\n\"iks20150304\"\r\n

      \\[\\begin{align}\r\nV _ a & = \\dfrac{1}{3} \\cdot 1^2 \\pi \\cdot 1 = \\dfrac{\\pi}{3} , \\\\\r\nV _ c & = \\dfrac{1}{3} \\cdot \\left( \\dfrac{1}{4} \\right)^2 \\pi \\cdot \\dfrac{3}{4} = \\dfrac{\\pi}{64} , \\\\\r\nV _ b & = \\pi \\displaystyle\\int _ 0^{\\frac{1}{4}} \\left( x -2 \\sqrt{x} +1 \\right)^2 \\, dx \\\\\r\n& = \\pi \\left[ \\dfrac{x^3}{3} -\\dfrac{8 x^{\\frac{5}{2}}}{5}+3 x^2 -\\dfrac{8 x^{\\frac{3}{2}}}{3} +x \\right] _ 0^{\\frac{1}{4}} \\\\\r\n& = \\dfrac{1}{3} \\cdot \\dfrac{1}{64} -\\dfrac{1}{20} +\\dfrac{3}{16} -\\dfrac{1}{3} +\\dfrac{1}{4} \\\\\r\n& = \\dfrac{1}{5} -\\dfrac{9}{64} = \\dfrac{19}{320} \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\\begin{align}\r\nV & = V _ a +V _ b +V _ c \\\\\r\n& = \\pi \\left( \\dfrac{19 +5}{320} +\\dfrac{1}{3} \\right) \\\\\r\n& = \\dfrac{9 +40}{120} \\pi = \\underline{\\dfrac{49 \\pi}{120}} \\ .\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u5ea7\u6a19\u5e73\u9762\u4e0a\u3067\u6b21\u306e\u3088\u3046\u306b\u5a92\u4ecb\u5909\u6570\u8868\u793a\u3055\u308c\u308b\u66f2\u7dda \\(C\\) \u3092\u8003\u3048\u308b. \\[ \\left\\{ \\begin{array}{l} x = | \\cos t | \\cos^3 t \\\\ y = | \\sin t | \\sin^ … \u7d9a\u304d\u3092\u8aad\u3080 →<\/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":[130],"tags":[145,137],"class_list":["post-1265","post","type-post","status-publish","format-standard","hentry","category-ikashika_2015","tag-ikashika","tag-137"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1265","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=1265"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1265\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1265"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1265"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1265"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}