{"id":1130,"date":"2015-06-23T01:41:58","date_gmt":"2015-06-22T16:41:58","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1130"},"modified":"2021-09-24T17:33:21","modified_gmt":"2021-09-24T08:33:21","slug":"tok201404","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok201404\/","title":{"rendered":"\u6771\u5de5\u59272014\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\u70b9 P \\((t,s)\\) \u304c \\(s = \\sqrt{2} t^2 -2t\\) \u3092\u6e80\u305f\u3057\u306a\u304c\u3089 \\(xy\\) \u5e73\u9762\u4e0a\u3092\u52d5\u304f\u3068\u304d, \u70b9 P \u3092\u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3057\u3066 \\(45^{\\circ}\\) \u56de\u8ee2\u3057\u305f\u70b9 Q \u306e\u8ecc\u8de1\u3068\u3057\u3066\u5f97\u3089\u308c\u308b\u66f2\u7dda\u3092 \\(C\\) \u3068\u3059\u308b.\r\n\u3000\u3055\u3089\u306b, \u66f2\u7dda \\(C\\) \u3068 \\(x\\) \u8ef8\u3067\u56f2\u307e\u308c\u305f\u56f3\u5f62\u3092 \\(D\\) \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u70b9 Q \\((x,y)\\) \u306e\u5ea7\u6a19\u3092, \\(t\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u76f4\u7dda \\(y = a\\) \u3068\u66f2\u7dda \\(C\\) \u304c\u305f\u3060 \\(1\\) \u3064\u5171\u6709\u70b9\u3092\u6301\u3064\u3088\u3046\u306a\u5b9a\u6570 \\(a\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u56f3\u5f62 \\(D\\) \u3092 \\(y\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3057\u3066\u5f97\u3089\u308c\u308b\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d \\(V\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

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

    Q \\(( X , Y )\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\n\\left( \\begin{array}{c} X \\\\ Y \\end{array} \\right) & = \\left( \\begin{array}{cc} \\cos 45^{\\circ} & -\\sin 45^{\\circ} \\\\ \\sin 45^{\\circ} & \\cos 45^{\\circ} \\end{array} \\right) \\left( \\begin{array}{c} t \\\\ s \\end{array} \\right) \\\\\r\n& = \\dfrac{\\sqrt{2}}{2} \\left( \\begin{array}{cc} 1 & -1 \\\\ 1 & 1 \\end{array} \\right) \\left( \\begin{array}{c} t \\\\ \\sqrt{2} t^2 -2t \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{c} -t^2 +\\dfrac{3 \\sqrt{2} t}{2} \\\\ t^2 -\\dfrac{\\sqrt{2} t}{2} \\end{array} \\right)\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u70b9 Q \u306e\u5ea7\u6a19\u306f\r\n\\[\r\n\\underline{\\left( -t^2 +\\dfrac{3 \\sqrt{2} t}{2} , \\ t^2 -\\dfrac{\\sqrt{2} t}{2} \\right)}\r\n\\]\r\n

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

    \\(t\\) \u306b\u95a2\u3059\u308b \\(2\\) \u6b21\u65b9\u7a0b\u5f0f \\(Y=a\\) \u304c \\(1\\) \u3064\u3060\u3051\u3082\u3064\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.\r\n\\[\\begin{align}\r\nt^2 -\\dfrac{\\sqrt{2} t}{2} & = a \\\\\r\n\\sqrt{2} t^2 -t -\\sqrt{2} a & = 0\r\n\\end{align}\\]\r\n\u3053\u306e\u65b9\u7a0b\u5f0f\u306e\u5224\u5225\u5f0f\u3092 \\(D\\) \u3068\u3059\u308c\u3070, \u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\r\nD = 1^2 -4 \\cdot \\sqrt{2} \\left( -\\sqrt{2} a \\right) = 0 \\\\\r\n\\text{\u2234} \\quad a = \\underline{-\\dfrac{1}{8}}\r\n\\]\r\n

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

    \\(D\\) \u3068 \\(x\\) \u8ef8\u304c\u4ea4\u308f\u308b\u3068\u304d\u306e \\(t\\) \u306e\u5024\u3092\u6c42\u3081\u308b\u3068\r\n\\[\\begin{align}\r\nt^2 -\\dfrac{\\sqrt{2} t}{2} & = 0 \\\\\r\nt \\left( \\sqrt{2} t -1 \\right) & = 0 \\\\\r\n\\text{\u2234} \\quad t = 0 , & \\ \\dfrac{\\sqrt{2}}{2}\r\n\\end{align}\\]\r\n\\(t = 0\\) \u306e\u3068\u304d, \\(X = 0\\) . \\(t = \\dfrac{\\sqrt{2}}{2}\\) \u306e\u3068\u304d, \\(X = 1\\) .
    \r\n\u307e\u305f\r\n\\[\r\n\\dfrac{dX}{dt} = -2t +\\dfrac{3 \\sqrt{2}}{2}\r\n\\]\r\n\"tok20140401\"\r\n

    \u56de\u8ee2\u3055\u305b\u308b\u9818\u57df\u304c \\(y \\lt 0\\) \u306b\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308c\u3070, \u6c42\u3081\u308b\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\\begin{align}\r\nV & = \\displaystyle\\int _ {0}^{1} 2 \\pi X (-Y) \\, dX \\\\\r\n& = 2 \\pi \\displaystyle\\int _ {0}^{\\frac{\\sqrt{2}}{2}} \\left( -t^2 +\\dfrac{3 \\sqrt{2} t}{2} \\right) \\left( -t^2 +\\dfrac{\\sqrt{2} t}{2} \\right) \\left( -2t +\\dfrac{3 \\sqrt{2}}{2} \\right) \\, dt \\\\\r\n& = \\dfrac{\\sqrt{2} \\pi}{2} \\displaystyle\\int _ {0}^{\\frac{\\sqrt{2}}{2}} t^2 \\left( -\\sqrt{2} t +3 \\right) \\left( -\\sqrt{2} t +1 \\right) \\left( -2 \\sqrt{2} t +3 \\right) \\, dt \\\\\r\n& = \\dfrac{\\sqrt{2} \\pi}{2} \\displaystyle\\int _ {0}^{\\frac{\\sqrt{2}}{2}} \\left( -4 \\sqrt{2} t^5 +22 t^4 -18 \\sqrt{2} t^3 +9 t^2 \\right) \\, dt \\\\\r\n& = \\dfrac{\\sqrt{2} \\pi}{2} \\left[ -\\dfrac{2 \\sqrt{2} t^6}{3} +\\dfrac{22 t^5}{5} -\\dfrac{18 \\sqrt{2} t^4}{4} +3 t^3 \\right] _ {0}^{\\frac{\\sqrt{2}}{2}} \\\\\r\n& = \\pi \\left( -\\dfrac{1}{12} +\\dfrac{11}{20} -\\dfrac{9}{8} +\\dfrac{3}{4} \\right) \\\\\r\n& = \\underline{\\dfrac{11 \\pi}{120}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u70b9 P \\((t,s)\\) \u304c \\(s = \\sqrt{2} t^2 -2t\\) \u3092\u6e80\u305f\u3057\u306a\u304c\u3089 \\(xy\\) \u5e73\u9762\u4e0a\u3092\u52d5\u304f\u3068\u304d, \u70b9 P \u3092\u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3057\u3066 \\(45^{\\circ}\\) \u56de\u8ee2\u3057\u305f\u70b9 Q \u306e\u8ecc\u8de1\u3068\u3057\u3066\u5f97 … \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":[116],"tags":[141,112],"class_list":["post-1130","post","type-post","status-publish","format-standard","hentry","category-toko_2014","tag-toko","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1130","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=1130"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1130\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1130"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1130"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1130"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}