{"id":1255,"date":"2015-08-27T10:01:08","date_gmt":"2015-08-27T01:01:08","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1255"},"modified":"2021-09-08T20:47:16","modified_gmt":"2021-09-08T11:47:16","slug":"osr201504","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/osr201504\/","title":{"rendered":"\u962a\u5927\u7406\u7cfb2015\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\u5ea7\u6a19\u7a7a\u9593\u306e \\(x\\) \u8ef8\u4e0a\u306b\u52d5\u70b9 P , Q \u304c\u3042\u308b.\r\nP , Q \u306f\u6642\u523b \\(0\\) \u306b\u304a\u3044\u3066, \u539f\u70b9\u3092\u51fa\u767a\u3059\u308b.\r\nP \u306f \\(x\\) \u8ef8\u306e\u6b63\u306e\u65b9\u5411\u306b, Q \u306f \\(x\\) \u8ef8\u306e\u8ca0\u306e\u65b9\u5411\u306b, \u3068\u3082\u306b\u901f\u3055 \\(1\\) \u3067\u52d5\u304f.\r\n\u305d\u306e\u5f8c, \u3068\u3082\u306b\u6642\u523b \\(1\\) \u3067\u505c\u6b62\u3059\u308b.\r\n\u70b9 P , Q \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(1\\) \u306e\u7403\u3092\u305d\u308c\u305e\u308c \\(A , B\\) \u3068\u3057, \u7a7a\u9593\u3067 \\(x \\geqq -1\\) \u306e\u90e8\u5206\u3092 \\(C\\) \u3068\u3059\u308b.\r\n\u3053\u306e\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u6642\u523b \\(t \\ ( 0 \\leqq t \\leqq 1 )\\) \u306b\u304a\u3051\u308b\u7acb\u4f53 \\(( A \\cup B ) \\cap C\\) \u306e\u4f53\u7a4d \\(V(t)\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\\(V(t)\\) \u306e\u6700\u5927\u5024\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

    \u5bfe\u79f0\u6027\u304b\u3089, \\(A , B\\) \u306e\u7403\u9762\u306e\u4ea4\u308f\u308a\u306f, \u5e73\u9762 \\(x = 0\\) \u4e0a\u306b\u3042\u308b.<\/p>\r\n\"osr20150401\"\r\n

    \u4e0a\u56f3\u306e\u3088\u3046\u306b \\(xy\\) \u5e73\u9762\u3092\u8a2d\u5b9a\u3057, \\(2\\) \u3064\u306e\u534a\u5186 \\(A' , B'\\) \u306e\u5f0f\u3092\u305d\u308c\u305e\u308c \\(y _ A , y _ B\\) \u3068\u304a\u3051\u3070\r\n\\[\\begin{align}\r\ny _ A & = \\sqrt{1 -(x-t)^2} \\\\\r\ny _ B & = \\sqrt{1 -(x+t)^2} \\ .\r\n\\end{align}\\]\r\n\u7acb\u4f53 \\(( A \\cup B ) \\cap C\\) \u306f, \u4e0a\u56f3\u659c\u7dda\u90e8\u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b\u56de\u8ee2\u3055\u305b\u305f\u56de\u8ee2\u4f53\u306b\u76f8\u5f53\u3059\u308b.
    \r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u4f53\u7a4d\u306f\r\n\\[\\begin{align}\r\nV(t) & = \\pi \\displaystyle\\int _ {-1}^0 {y _ B}^2 \\, dx +\\pi \\displaystyle\\int _ 0^{1+t} {y _ A}^2 \\, dx \\\\\r\n& = \\pi \\displaystyle\\int _ {-1}^0 \\left\\{ 1 -(x+t)^2 \\right\\} \\, dx +\\pi \\displaystyle\\int _ 0^{1+t} \\left\\{ 1 -(x-t)^2 \\right\\} \\, dx \\\\\r\n& = \\pi \\left[ x -\\dfrac{(x+t)^3}{3} \\right] _ {-1}^0 +\\pi \\left[ x -\\dfrac{(x-t)^3}{3} \\right] _ 0^{1+t} \\\\\r\n& = \\pi \\left\\{ -\\dfrac{t^3}{3} +1 +\\dfrac{(t-1)^3}{3} \\right\\} +\\pi \\left( 1+t -\\dfrac{1}{3} -\\dfrac{t^3}{3} \\right) \\\\\r\n& = \\underline{\\dfrac{\\pi}{3} ( -t^3 -3t^2 +6t +4 )} \\ .\r\n\\end{align}\\]\r\n

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

    \\(f(t) = -t^3 -3t^2 +6t +4\\) \u3068\u304a\u3051\u3070, \\(f(t)\\) \u304c\u6700\u5927\u306b\u306a\u308b\u3068\u304d, \\(V(t)\\) \u3082\u6700\u5927\u306b\u306a\u308b.
    \r\n\\[\\begin{align}\r\nf'(t) & = -3t^2 -6t +6 \\\\\r\n& = -3 ( t^2 +2t -2 ) \\ .\r\n\\end{align}\\]\r\n\\(f'(t) = 0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\nt = -1 +\\sqrt{1+2} = \\sqrt{3} -1 \\ .\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(0 \\leqq t \\leqq 1\\) \u306b\u304a\u3051\u308b \\(f(t)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} t & 0 & \\cdots & \\sqrt{3} -1 & \\cdots & 1 \\\\ \\hline f'(t) & & + & 0 & - & \\\\ \\hline f(t) & & \\nearrow & \\text{\u6700\u5927} & \\searrow & \\end{array}\r\n\\]\r\n\\(f(x = -(x+1) (t^2+2t-2) +6t+2\\) \u3067\u3042\u308b\u3053\u3068\u3092\u7528\u3044\u308c\u3070\r\n\\[\r\nf \\left( \\sqrt{3} -1 \\right) = 6 \\sqrt{3} -4 \\ .\r\n\\]\r\n\u3088\u3063\u3066, \\(V(t)\\) \u306e\u6700\u5927\u5024\u306f\r\n\\[\r\n\\dfrac{\\pi}{3} f \\left( \\sqrt{3} -1 \\right) = \\underline{\\left( 2 \\sqrt{3} -\\dfrac{4}{3} \\right) \\pi} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u5ea7\u6a19\u7a7a\u9593\u306e \\(x\\) \u8ef8\u4e0a\u306b\u52d5\u70b9 P , Q \u304c\u3042\u308b. P , Q \u306f\u6642\u523b \\(0\\) \u306b\u304a\u3044\u3066, \u539f\u70b9\u3092\u51fa\u767a\u3059\u308b. P \u306f \\(x\\) \u8ef8\u306e\u6b63\u306e\u65b9\u5411\u306b, Q \u306f \\(x\\) \u8ef8\u306e\u8ca0\u306e\u65b9\u5411\u306b, \u3068\u3082\u306b\u901f\u3055 \\(1\\) … \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":[127],"tags":[142,137],"class_list":["post-1255","post","type-post","status-publish","format-standard","hentry","category-osaka_r_2015","tag-osaka_r","tag-137"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1255","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=1255"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1255\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1255"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1255"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1255"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}