{"id":1981,"date":"2021-11-18T08:49:45","date_gmt":"2021-11-17T23:49:45","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1981"},"modified":"2021-11-20T15:10:57","modified_gmt":"2021-11-20T06:10:57","slug":"tok202105","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok202105\/","title":{"rendered":"\u6771\u5de5\u59272021\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\\(xy\\) \u5e73\u9762\u4e0a\u306e\u5186 \\(C : x^2 +(y-a)^2 = a^2 \\ ( a \\gt 0 )\\) \u3092\u8003\u3048\u308b.\r\n\u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u5186 \\(C\\) \u304c \\(y \\geqq x^2\\) \u3067\u8868\u3055\u308c\u308b\u9818\u57df\u306b\u542b\u307e\u308c\u308b\u305f\u3081\u306e \\(a\\) \u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u5186 \\(C\\) \u304c \\(y \\geqq x^2 -x^4\\) \u3067\u8868\u3055\u308c\u308b\u9818\u57df\u306b\u542b\u307e\u308c\u308b\u305f\u3081\u306e \\(a\\) \u306e\u7bc4\u56f2\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\\(a\\) \u304c (2)<\/strong> \u306e\u7bc4\u56f2\u306b\u3042\u308b\u3068\u3059\u308b.\r\n\\(xy\\) \u5e73\u9762\u306b\u304a\u3044\u3066\u9023\u7acb\u4e0d\u7b49\u5f0f\r\n\\[\r\n| x | \\leqq \\dfrac{1}{\\sqrt{2}} \\ , \\ 0 \\leqq y \\leqq \\dfrac{1}{4} \\ , \\ y \\geqq x^2 -x^4 \\ , \\ x^2 +(y-a)^2 \\geqq a^2\r\n\\]\r\n\u3067\u8868\u3055\u308c\u308b\u9818\u57df \\(D\\) \u3092, \\(y\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3055\u305b\u3066\u3067\u304d\u308b\u7acb\u4f53\u306e\u4f53\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n

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

    \u70b9 \\(( t , t^2 )\\) \u304c\u5e38\u306b \\(C\\) \u306e\u5916\u5074\uff08\u5883\u754c\u542b\u3080\uff09\u306b\u3042\u308b\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044\u306e\u3067\r\n\\[\\begin{align}\r\nt^2 +( t^2 -a )^2 & \\geqq a^2 \\\\\r\nt^2 \\left( t^2 -2a +1 \\right) & \\geqq 0 \\\\\r\n\\text{\u2234} \\quad t^2 -2a +1 & \\geqq 0 \\quad ( \\ \\text{\u2235} \\ t^2 \\geqq 0 \\ )\r\n\\end{align}\\]\r\n\u3053\u308c\u304c\u5e38\u306b\u6210\u7acb\u3059\u308b\u6761\u4ef6\u306f\r\n\\[\\begin{gather}\r\n-2a +1 \\geqq 0 \\\\\r\n\\text{\u2234} \\quad \\underline{0 \\lt a \\leqq \\dfrac{1}{2}}\r\n\\end{gather}\\]\r\n

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

    \\(x^2 -x^4 \\leqq x^2\\) \u306a\u306e\u3067, \u70b9 \\(( t , t^2 -t^4 )\\) \u306f, \u70b9 \\(( t , t^2 )\\) \u306e\u5e38\u306b\u4e0b\u5074\u306b\u3042\u308a, \u70b9 \\(( 0 , a )\\) \u3068\u306e\u8ddd\u96e2\u304c\u5927\u304d\u3044.
    \r\n\u3086\u3048\u306b, (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a, \\(0 \\lt a \\leqq \\dfrac{1}{2}\\) \u306f\u6761\u4ef6\u3092\u307f\u305f\u3057\u3066\u3044\u308b.
    \r\n\u4ee5\u4e0b\u3067\u306f, \\(a \\gt \\dfrac{1}{2}\\) \u306e\u3068\u304d\u306b\u3064\u3044\u3066\u8003\u3048\u308b.
    \r\n\u70b9 \\(( t , t^2 -t^4 )\\) \u304c\u5e38\u306b \\(C\\) \u306e\u5916\u5074\uff08\u5883\u754c\u542b\u3080\uff09\u306b\u3042\u308b\u6761\u4ef6\u3092\u6c42\u3081\u308c\u3070\u3088\u3044.
    \r\n\\[\\begin{align}\r\nt^2 +( t^2 -t^4 -a )^2 & \\geqq a^2 \\\\\r\nt^2 \\left\\{ t^6 -2t^4 +(2a+1) t^2 -2a +1 \\right\\} & \\geqq 0 \\\\\r\n\\text{\u2234} \\quad t^6 -2t^4 +(2a+1) t^2 -2a +1 & \\geqq 0 \\quad ( \\ \\text{\u2235} \\ t^2 \\geqq 0 \\ )\r\n\\end{align}\\]\r\n\u3057\u304b\u3057, \\(t = 0\\) \u306e\u3068\u304d\r\n\\[\r\n( \\text{\u5de6\u8fba} ) = -2a +1 \\lt 0\r\n\\]\r\n\u306a\u306e\u3067, \u4e0d\u7b49\u5f0f\u306f\u5e38\u306b\u306f\u6210\u7acb\u3057\u306a\u3044.
    \r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\r\n\\underline{0 \\lt a \\leqq \\dfrac{1}{2}}\r\n\\]\r\n

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

    \u9818\u57df \\(D\\) \u306f \\(y\\) \u8ef8\u306b\u3064\u3044\u3066\u5bfe\u8c61\u306a\u306e\u3067, \\(x \\geqq 0\\) \u306e\u90e8\u5206\u306b\u3064\u3044\u3066\u8003\u3048\u308c\u3070\u3088\u3044.
    \r\n\\(y = x^2 -x^4\\) \u3088\u308a\r\n\\[\r\ny' = 2x -4x^3 = -2x ( 2 x^2 -1 )\r\n\\]\r\n\u306a\u306e\u3067, \\(|x| \\leqq \\dfrac{1}{\\sqrt{2}} , \\ 0 \\leqq y \\leqq \\dfrac{1}{4} , \\ y \\geqq x^2 -x^4\\) \u306e\u793a\u3059\u9818\u57df \\(D_1\\) \u306f, \u4e0b\u56f3\u659c\u7dda\u90e8\u3068\u306a\u308b.<\/p>\r\n\"tok20210501\"\r\n

    \u3053\u308c\u306e \\(y\\) \u8ef8\u306b\u3088\u308b\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d \\(V_1\\) \u306f\r\n\\[\\begin{align}\r\n\uff36_1 & = 2 \\pi \\displaystyle\\int _ {0}^{\\frac{1}{\\sqrt{2}}} x \\left( \\dfrac{1}{4} -x^2 +x^4 \\right) \\, dx \\\\\r\n& = 2 \\pi \\left[ \\dfrac{x^2}{16} -\\dfrac{x^4}{4} +\\dfrac{x^6}{6} \\right] _ {0}^{\\frac{1}{\\sqrt{2}}} \\\\\r\n& = 2\\pi \\left( \\dfrac{1}{16} -\\dfrac{1}{16} +\\dfrac{\\pi}{48} \\right) = \\dfrac{\\pi}{24}\r\n\\end{align}\\]\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(0 \\lt a \\lt \\dfrac{1}{8}\\) \u306e\u3068\u304d
      \r\n\\(C\\) \u5168\u4f53\u304c \\(y = \\dfrac{1}{4}\\) \u306e\u4e0b\u5074\u306b\u3042\u308a, \\(C\\) \u306e \\(y\\) \u8ef8\u306b\u3088\u308b\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d \\(V_2\\) \u306f\r\n\\[\r\nV_2 = \\dfrac{4 a^3 \\pi}{3}\r\n\\]\r\n\u3086\u3048\u306b, \u6c42\u3081\u308b\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\r\nV = V_1 -V_2 = \\dfrac{\\pi}{24} -\\dfrac{4 a^3 \\pi}{3}\r\n\\]<\/li>\r\n

    2. 2*<\/strong>\u3000\\(\\dfrac{1}{8} \\leqq a \\leqq \\dfrac{1}{2}\\) \u306e\u3068\u304d
      \r\n\\(C\\) \u306e\u4e00\u90e8\u304c \\(y = \\dfrac{1}{4}\\) \u306e\u4e0b\u5074\u306b\u3042\u308a, \u3053\u306e\u90e8\u5206\u306e \\(y\\) \u8ef8\u306b\u3088\u308b\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d \\(V_3\\) \u306f\r\n\\[\\begin{align}\r\nV_3 & = \\pi \\displaystyle\\int _ {0}^{\\frac{1}{4}} ( 2ay -y^2 ) \\, dy \\\\\r\n& = \\pi \\left[ ay^2 -\\dfrac{y^3}{3} \\right] _ {0}^{\\frac{1}{4}} \\\\\r\n& = \\dfrac{a \\pi}{16} -\\dfrac{\\pi}{192}\r\n\\end{align}\\]\r\n\u3086\u3048\u306b, \u6c42\u3081\u308b\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\r\nV = V_1 -V_3 = \\dfrac{3 \\pi}{64} -\\dfrac{a \\pi}{16}\r\n\\]<\/li>\r\n<\/ol>\r\n

      \u4ee5\u4e0a\u3088\u308a\r\n\\[\r\nV = \\underline{\\left\\{ \\begin{array}{ll} \\dfrac{\\pi}{24} -\\dfrac{4 a^3 \\pi}{3} & \\left( \\ 0 \\lt a \\lt \\dfrac{1}{8} \\text{\u306e\u3068\u304d}\\ \\right) \\\\ \\dfrac{3 \\pi}{64} -\\dfrac{a \\pi}{16} & \\left( \\ \\dfrac{1}{8} \\leqq a \\leqq \\dfrac{1}{2} \\text{\u306e\u3068\u304d}\\ \\right) \\end{array} \\right.}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u4e0a\u306e\u5186 \\(C : x^2 +(y-a)^2 = a^2 \\ ( a \\gt 0 )\\) \u3092\u8003\u3048\u308b. \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088. (1)\u3000\u5186 \\(C\\) \u304c \\(y \\geqq x^2\\) \u3067\u8868\u3055\u308c\u308b\u9818\u57df\u306b\u542b … \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":[173],"tags":[141,165],"class_list":["post-1981","post","type-post","status-publish","format-standard","hentry","category-toko_2021","tag-toko","tag-165"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1981","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=1981"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1981\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1981"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1981"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1981"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}