{"id":402,"date":"2012-05-23T23:55:02","date_gmt":"2012-05-23T14:55:02","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=402"},"modified":"2021-09-14T08:38:29","modified_gmt":"2021-09-13T23:38:29","slug":"iks201202","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/iks201202\/","title":{"rendered":"\u533b\u79d1\u6b6f\u79d1\u59272012\uff1a\u7b2c2\u554f"},"content":{"rendered":"
\n

\\(a^2+b^2=1\\) \u3092\u6e80\u305f\u3059\u6b63\u306e\u5b9f\u6570 \\(a , b\\) \u306e\u5168\u4f53\u3092 \\(S\\) \u3068\u3059\u308b.\r\n\\(S\\) \u306b\u542b\u307e\u308c\u308b \\((a, b)\\) \u306b\u5bfe\u3057, \\(xyz\\) \u7a7a\u9593\u5185\u306b \\(3\\) \u70b9 P \\((a, b, b)\\) , Q \\((-a, b, b)\\) , R \\((0, 0, b)\\) \u3092\u3068\u308b. \u307e\u305f\u539f\u70b9\u3092 O \u3068\u3059\u308b. \u3053\u306e\u3068\u304d\u4ee5\u4e0b\u306e\u5404\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u4e09\u89d2\u5f62 OPQ \u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u7acb\u4f53\u3092 \\(F _ 1\\) \u3068\u3059\u308b. \\((a, b)\\) \u304c \\(S\\) \u306e\u4e2d\u3092\u52d5\u304f\u3068\u304d, \\(F _ 1\\) \u306e\u4f53\u7a4d\u306e\u6700\u5927\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u4e09\u89d2\u5f62 PQR \u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u7acb\u4f53\u3092 \\(F _ 2\\) \u3068\u3059\u308b. \\(a = b = \\dfrac{1}{\\sqrt{2}}\\) \u306e\u3068\u304d, \\(F _ 2\\) \u306e \\(xy\\) \u5e73\u9762\u306b\u3088\u308b\u5207\u308a\u53e3\u306e\u5468\u3092 \\(xy\\) \u5e73\u9762\u4e0a\u306b\u56f3\u793a\u305b\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u4e09\u89d2\u5f62 OPR \u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u7acb\u4f53\u3092 \\(F _ 3\\) \u3068\u3059\u308b. \\((a, b)\\) \u304c \\(S\\) \u306e\u4e2d\u3092\u52d5\u304f\u3068\u304d, \\(F _ 3\\) \u306e\u4f53\u7a4d\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

    \u4e09\u89d2\u5f62 OPQ \u306f \\(x\\) \u8ef8\u3068\u5782\u76f4\u3067\u3042\u308b.
    \r\nPQ \u306e\u4e2d\u70b9\u3092 M \u3068\u304a\u3051\u3070\r\n\\[\r\n\\text{PQ} =2a , \\ \\text{OM} =\\sqrt{2} b\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(F _ 1\\) \u306e\u4f53\u7a4d \\(V _ 1\\) \u306f\r\n\\[\\begin{align}\r\nV _ 1 & = 2b^2 \\pi \\cdot 2a -2 \\cdot \\dfrac{1}{3} \\cdot 2b^2 \\pi \\cdot a \\\\\r\n& = \\dfrac{8 \\pi}{3} ab^2 \\\\\r\n& = \\dfrac{8 \\pi}{3} \\underline{a(1-a^2)} _ {[1]}\n\\end{align}\\]\r\n\u4e0b\u7dda\u90e8 [1] \u304c\u6700\u5927\u3068\u306a\u308b\u3068\u304d, \\(V _ 1\\) \u3082\u6700\u5927\u3068\u306a\u308b.
    \r\n[1] \u3092 \\(f(a)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nf'(a) =1 -3a^2 = -\\left( \\sqrt{3} a+1 \\right) \\left( \\sqrt{3} a-1 \\right)\n\\]\r\n\\(f'(a)=0\\) \u3092\u3068\u304f\u3068\r\n\\[\r\na = \\dfrac{\\sqrt{3}}{3}\n\\]\r\n\u306a\u306e\u3067, \\(0 \\lt a \\lt 1\\) \u306b\u304a\u3051\u308b \\(f(a)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3068\u304a\u308a.\r\n\\[\r\n\\begin{array}{c|ccccc} a & (0) & \\cdots & \\frac{\\sqrt{3}}{3} & \\cdots & (1) \\\\ \\hline f'(a) & & + & 0 & - & \\\\ \\hline f(a) & & \\nearrow & \\text{\u6700\u5927} & \\searrow \\end{array}\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\r\nf \\left( \\dfrac{\\sqrt{3}}{3} \\right) = \\dfrac{2 \\sqrt{3}}{9}\n\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6700\u5927\u5024\u306f\r\n\\[\r\n\\dfrac{8 \\pi}{3} f \\left( \\dfrac{\\sqrt{3}}{3} \\right) = \\underline{\\dfrac{16 \\sqrt{3} \\pi}{27}}\n\\]\r\n

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

    \u5bfe\u79f0\u6027\u304b\u3089, \\(x \\geqq 0 , y \\geqq 0\\) \u306e\u90e8\u5206\u306b\u3064\u3044\u3066\u8003\u3048\u308b.
    \r\nPQ , PR \u3068 \\(x = t \\ \\left( 0 \\leqq t \\leqq \\dfrac{1}{2} \\right)\\) \u5e73\u9762\u3068\u306e\u4ea4\u70b9\u3092 S , T \u3068\u304a\u304f\u3068\r\n\\[\r\n\\text{S} \\ \\left( t , t , \\dfrac{\\sqrt{2}}{2} \\right) , \\quad \\text{T} \\ \\left( t , \\dfrac{\\sqrt{2}}{2} , \\dfrac{\\sqrt{2}}{2} \\right)\n\\]\r\n\\(F _ 2\\) \u306e \\(x = t\\) \u5e73\u9762\u306b\u304a\u3051\u308b\u65ad\u9762\u306f, S , T \u306e\u66f8\u304f\u5186\u5468\u306b\u56f2\u307e\u308c\u305f\u90e8\u5206\u306b\u306a\u308b.
    \r\n\u305d\u308c\u305e\u308c\u306e\u534a\u5f84\u3092 \\(r _ 2 , R _ 2\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nr _ 2 = \\sqrt{t^2 +\\dfrac{1}{2}} , \\ R _ 2 = 1\n\\]\r\n\u3088\u3063\u3066, \\(F _ 2\\) \u306e\u65ad\u9762\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\u3068\u306a\u308b.<\/p>\r\n\"\"\r\n

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

    OP , PR \u3068 \\(x = t \\ ( 0 \\leqq t \\leqq a )\\) \u5e73\u9762\u3068\u306e\u4ea4\u70b9\u3092 U , V \u3068\u304a\u304f\u3068\r\n\\[\r\n\\text{U} \\ \\left( t , t , \\dfrac{bt}{a} \\right) , \\quad \\text{V} \\ \\left( t , t , b \\right)\n\\]\r\n\\(F _ 3\\) \u306e \\(x = t\\) \u5e73\u9762\u306b\u304a\u3051\u308b\u65ad\u9762\u306f, U , V \u306e\u66f8\u304f\u5186\u5468\u306b\u56f2\u307e\u308c\u305f\u90e8\u5206\u306b\u306a\u308b.
    \r\n\u3053\u306e\u90e8\u5206\u306e\u9762\u7a4d\u3092 \\(S(t)\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nS(t) & = \\left( t^2 +b^2 \\right) -\\left\\{ t^2 +\\left( \\dfrac{bt}{a} \\right)^2 \\right\\} \\\\\r\n& =b^2 \\left( 1 -\\dfrac{t^2}{a^2} \\right)\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(F _ 3\\) \u306e\u4f53\u7a4d \\(V _ 3\\) \u306f\r\n\\[\\begin{align}\r\nV _ 3 & = \\pi \\displaystyle\\int _ 0^a S(t) \\, dt = \\pi b^2 \\left[ t -\\dfrac{t^3}{3 a^2} \\right] _ 0^a \\\\\r\n& = \\dfrac{2 \\pi ab^2}{3} = \\dfrac{2 \\pi}{3} a(1-a^2)\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(V _ 3\\) \u304c\u6700\u5927\u3068\u306a\u308b\u306e\u306f (1)<\/strong> \u3068\u540c\u69d8\u306e\u3068\u304d\u3067, \u6c42\u3081\u308b\u6700\u5927\u5024\u306f\r\n\\[\r\n\\dfrac{2 \\pi}{3} f \\left( \\dfrac{\\sqrt{3}}{3} \\right) = \\underline{\\dfrac{4 \\sqrt{3} \\pi}{27}}\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a^2+b^2=1\\) \u3092\u6e80\u305f\u3059\u6b63\u306e\u5b9f\u6570 \\(a , b\\) \u306e\u5168\u4f53\u3092 \\(S\\) \u3068\u3059\u308b. \\(S\\) \u306b\u542b\u307e\u308c\u308b \\((a, b)\\) \u306b\u5bfe\u3057, \\(xyz\\) \u7a7a\u9593\u5185\u306b \\(3\\) \u70b9 P \\((a, b, b … \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":[62],"tags":[145,68],"class_list":["post-402","post","type-post","status-publish","format-standard","hentry","category-ikashika_2012","tag-ikashika","tag-68"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/402","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=402"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/402\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=402"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=402"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=402"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}