{"id":214,"date":"2011-12-03T00:08:39","date_gmt":"2011-12-02T15:08:39","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=214"},"modified":"2021-11-04T13:11:40","modified_gmt":"2021-11-04T04:11:40","slug":"htb200902","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/htb200902\/","title":{"rendered":"\u4e00\u6a4b\u59272009\uff1a\u7b2c2\u554f"},"content":{"rendered":"
    \r\n

  1. (1)<\/strong>\u3000\u4efb\u610f\u306e\u89d2 \\(\\theta\\) \u306b\u5bfe\u3057\u3066, \\(-2 \\leqq x \\cos \\theta +y \\sin \\theta \\leqq y+1\\) \u304c\u6210\u7acb\u3059\u308b\u3088\u3046\u306a\u70b9 \\((x, y)\\) \u306e\u5168\u4f53\u304b\u3089\u306a\u308b\u9818\u57df\u3092 \\(xy\\) \u5e73\u9762\u4e0a\u306b\u56f3\u793a\u3057, \u305d\u306e\u9762\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u4efb\u610f\u306e\u89d2 \\(\\alpha , \\beta\\) \u306b\u5bfe\u3057\u3066, \\(-1 \\leqq x^2 \\cos \\alpha +y \\sin \\beta \\leqq 1\\) \u304c\u6210\u7acb\u3059\u308b\u3088\u3046\u306a\u70b9 \\((x, y)\\) \u306e\u5168\u4f53\u304b\u3089\u306a\u308b\u9818\u57df\u3092 \\(xy\\) \u5e73\u9762\u4e0a\u306b\u56f3\u793a\u3057, \u305d\u306e\u9762\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

    \u660e\u3089\u304b\u306b\r\n\\[\\begin{align}\r\n-2 & \\leqq y+1 \\\\\r\n\\text{\u2234} \\quad y & \\geqq -3\r\n\\end{align}\\]\r\n\u4e0e\u5f0f\u3092\u5909\u5f62\u3059\u308b\u3068\r\n\\[\r\n-2 \\leqq \\sqrt{x^2+y^2} \\cos ( \\theta +\\gamma ) \\leqq y+1\r\n\\]\r\n\u305f\u3060\u3057, \\(\\cos \\gamma = \\dfrac{x}{\\sqrt{x^2+y^2}}\\) , \\(\\sin \\gamma = -\\dfrac{y}{\\sqrt{x^2+y^2}}\\) .
    \r\n\u4e2d\u8fba\u3092 \\(A\\) \u3068\u304a\u3051\u3070, \\(\\sqrt{x^2+y^2} \\leqq A \\leqq -\\sqrt{x^2+y^2}\\) \u306a\u306e\u3067, \\(2\\) \u3068 \\(y+1\\) \u306e\u5927\u5c0f\u306b\u3088\u3063\u3066\u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(y+1 \\leqq 2\\) \u3059\u306a\u308f\u3061 \\(-3 \\leqq y \\leqq 1\\) \u306e\u3068\u304d
      \r\n\u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\\begin{align}\r\n\\sqrt{x^2+y^2} & \\leqq y+1 \\\\\r\n\\text{\u2234} \\quad x^2+y^2 & \\leqq (y+1)^2 \\\\\r\n\\text{\u2234} \\quad y & \\leqq \\dfrac{x^2-1}{2}\r\n\\end{align}\\]<\/li>\r\n

    2. 2*<\/strong>\u3000\\(y+1 \\gt 2\\) \u3059\u306a\u308f\u3061 \\(y \\gt 1\\) \u306e\u3068\u304d
      \r\n\u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\\begin{align}\r\n\\sqrt{x^2+y^2} & \\leqq 2 \\\\\r\n\\text{\u2234} \\quad x^2+y^2 & \\leqq 4\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n

      \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u9818\u57df\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u5883\u754c\u3092\u542b\u3080\uff09\u3068\u306a\u308b.<\/p>\r\n\"hitotsubashi_200902_01\"\r\n

      \u307e\u305f, \u3053\u306e\u9818\u57df\u306e\u9762\u7a4d \\(S\\) \u306f, \\(y = 1\\) \u306e\u4e0a\u4e0b\u306b\u5206\u3051\u3066\u8003\u3048\u3066\r\n\\[\\begin{align}\r\nS & = \\displaystyle\\int _ {-\\sqrt{3}}^{\\sqrt{3}} \\left( 1-\\dfrac{x^2-1}{2} \\right) \\, dx \\\\\r\n& \\qquad +\\dfrac{1}{2} \\cdot 2^2 \\cdot \\dfrac{2\\pi}{3} -\\dfrac{1}{2} \\cdot 2\\sqrt{3} \\cdot 1 \\\\\r\n& = \\dfrac{1}{2} \\cdot \\dfrac{\\left( 2 \\sqrt{3} \\right)^3}{6} +\\dfrac{4\\pi}{3} -\\sqrt{3} \\\\\r\n& = \\underline{\\dfrac{4\\pi}{3} +\\sqrt{3}}\r\n\\end{align}\\]\r\n

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

      \u4e0e\u5f0f\u306e\u4e2d\u8fba\u3092 \\(B\\) \u3068\u304a\u304f.
      \r\n\\(-1 \\leqq \\cos \\alpha \\leqq 1\\) , \\(-1 \\leqq \\sin \\beta \\leqq 1\\) \u306a\u306e\u3067, \\(y\\) \u306e\u6b63\u8ca0\u306b\u3088\u3063\u3066\u5834\u5408\u5206\u3051\u3057\u3066\u8003\u3048\u308b.<\/p>\r\n

        \r\n

      1. 1*<\/strong>\u3000\\(y \\geqq 0\\) \u306e\u3068\u304d\r\n\\[\r\n-x^2 -y \\leqq B \\leqq x^2+y\r\n\\]\r\n\u306a\u306e\u3067, \u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\\begin{align}\r\n0 \\leqq x^2+y & \\leqq 1 \\\\\r\n\\text{\u2234} \\quad 0 \\leqq y & \\leqq 1-x^2\r\n\\end{align}\\]<\/li>\r\n

      2. 2*<\/strong>\u3000\\(y \\lt 0\\) \u306e\u3068\u304d\r\n\\[\r\n-x^2 +y \\leqq B \\leqq x^2-y\r\n\\]\r\n\u306a\u306e\u3067, \u6c42\u3081\u308b\u6761\u4ef6\u306f\r\n\\[\\begin{align}\r\n0 \\leqq x^2-y & \\leqq 1 \\\\\r\n\\text{\u2234} \\quad x^2-1 \\leqq y & \\lt 0\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n

        \u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u9818\u57df\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\uff08\u5883\u754c\u3092\u542b\u3080\uff09\u3068\u306a\u308b.<\/p>\r\n\"hitotsubashi_200902_02\"\r\n

        \u307e\u305f, \u3053\u306e\u9818\u57df\u306e\u9762\u7a4d \\(T\\) \u306f\r\n\\[\\begin{align}\r\nT & = 2 \\displaystyle\\int _ {-1}^{1} \\left( 1-x^2 \\right) \\, dx \\\\\r\n& = 2 \\cdot \\dfrac{2^2}{6} =\\underline{\\dfrac{8}{3}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"(1)\u3000\u4efb\u610f\u306e\u89d2 \\(\\theta\\) \u306b\u5bfe\u3057\u3066, \\(-2 \\leqq x \\cos \\theta +y \\sin \\theta \\leqq y+1\\) \u304c\u6210\u7acb\u3059\u308b\u3088\u3046\u306a\u70b9 \\((x, y)\\) \u306e\u5168\u4f53\u304b\u3089\u306a\u308b\u9818\u57df\u3092 … \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":[45],"tags":[146,15],"class_list":["post-214","post","type-post","status-publish","format-standard","hentry","category-hitotsubashi_2009","tag-hitotsubashi","tag-15"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/214","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=214"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/214\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=214"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=214"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=214"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}