{"id":1330,"date":"2015-11-09T01:08:43","date_gmt":"2015-11-08T16:08:43","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1330"},"modified":"2021-10-21T08:27:25","modified_gmt":"2021-10-20T23:27:25","slug":"wsr201505","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr201505\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52015\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\\(a \\gt 0\\) \u3068\u3059\u308b.\r\n\\(xy\\) \u5e73\u9762\u4e0a\u306b\u70b9 A \\(( -\\sqrt{2} a , 0 )\\) , B \\(( \\sqrt{2} a , 0 )\\) \u3092\u56fa\u5b9a\u3059\u308b. \u52d5\u70b9 P \\(( x , y )\\) \u306f\u6761\u4ef6 \\(\\text{AP} +\\text{BP} = 4a\\) \u3092\u307f\u305f\u3059\u3082\u306e\u3068\u3059\u308b. \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u70b9 P \u306e\u8ecc\u8de1\u3068\u3057\u3066\u5f97\u3089\u308c\u308b\u66f2\u7dda\u306e\u65b9\u7a0b\u5f0f\u3092\u6c42\u3081\u3088. \u305f\u3060\u3057, \u7b54\u306e\u307f\u3067\u3088\u3044.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000(1)<\/strong> \u306e\u66f2\u7dda\u306e \\(-\\sqrt{2} a \\leqq x \\leqq \\sqrt{2} a\\) \u306e\u90e8\u5206\u3068, \u76f4\u7dda \\(x = -\\sqrt{2} a\\) , \u76f4\u7dda \\(x = \\sqrt{2} a\\) \u3067\u56f2\u307e\u308c\u308b\u56f3\u5f62\u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u7acb\u4f53\u3092\u8003\u3048\u308b. \u3053\u306e\u7acb\u4f53\u306e\u4f53\u7a4d \\(V\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000(2)<\/strong> \u306e\u7acb\u4f53\u306e\u8868\u9762\u7a4d \\(S\\) \u3092\u6c42\u3081\u3088. \u3053\u3053\u3067, \\(y = f(x)\\) \u306e\u30b0\u30e9\u30d5\u306e \\(p \\leqq x \\leqq q\\) \u306e\u90e8\u5206\u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u66f2\u9762\u306e\u9762\u7a4d\u306f\r\n\\[\r\n2 \\pi \\displaystyle\\int _ p^q \\sqrt{\\{ f(x) \\}^2 +\\{ f(x) f'(x) \\}^2} \\, dx\r\n\\]\r\n\u3068\u3057\u3066\u8a08\u7b97\u3057\u3066\u3088\u3044.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

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

    (1)<\/strong><\/p>\r\n\"wsr20150501\"\r\n

    \u6761\u4ef6\u3088\u308a, P \u306e\u8ecc\u8de1\u306f A , B \u3092\u7126\u70b9\u3068\u3059\u308b\u6955\u5186\u3067\u3042\u308a, \u9577\u5f84\u306f\r\n\\[\r\n\\dfrac{4a}{2} = 2a\r\n\\]\r\n\u77ed\u5f84\u306f\r\n\\[\r\n\\sqrt{(2a)^2 -( \\sqrt{2} a )^2} = \\sqrt{2} a\r\n\\]\r\n\u3086\u3048\u306b, \u6c42\u3081\u308b\u5f0f\u306f\r\n\\[\\begin{align}\r\n\\dfrac{x^2}{( 2a )^2} +\\dfrac{y^2}{( \\sqrt{2} a )^2} & = 1 \\\\\r\n\\text{\u2234} \\quad \\underline{x^2 +2y^2 = 4a^2} & \\quad ... [1]\r\n\\end{align}\\]\r\n

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

    \u5bfe\u79f0\u6027\u304b\u3089, \\(x \\geqq 0\\) , \\(y \\geqq 0\\) \u306e\u9818\u57df\u3092\u56de\u8ee2\u3055\u305b\u308b\u5834\u5408\u3092\u8003\u3048\u308c\u3070\u3088\u3044.
    \r\n\u6c42\u3081\u308b\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\\begin{align}\r\nV & = 2 \\pi \\displaystyle\\int _ 0^{\\sqrt{2} a} \\dfrac{1}{2} ( 4a^2 -x^2 ) \\, dx \\\\\r\n& = \\pi \\left[ 4a^2 x -\\dfrac{x^3}{3} \\right] _ 0^{\\sqrt{2} a} \\\\\r\n& = \\pi \\left( 4 \\sqrt{2} a^3 -\\dfrac{2 \\sqrt{2}}{3} a^3 \\right) \\\\\r\n& = \\underline{\\dfrac{10 \\sqrt{2} \\pi}{3} a^3}\r\n\\end{align}\\]\r\n

    (3)<\/strong><\/p>\r\n[1] \u306b\u3064\u3044\u3066, \\(x\\) \u3067\u5fae\u5206\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n2x +4y y' & = 0 \\\\\r\n\\text{\u2234} \\quad y y' & = -\\dfrac{x}{2}\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070, \u6c42\u3081\u308b\u8868\u9762\u7a4d \\(S\\) \u306f\r\n\\[\\begin{align}\r\nS & = 2 \\cdot 2 \\pi \\displaystyle\\int _ 0^{\\sqrt{2} a} \\sqrt{\\dfrac{1}{2} \\left( 4a^2 -x^2 \\right) +\\left( -\\dfrac{x}{2} \\right)^2} \\, dx +2 \\cdot a^2 \\pi \\\\\r\n& = 2 \\pi \\underline{\\displaystyle\\int _ 0^{\\sqrt{2} a} \\sqrt{8a^2 -x^2} \\, dx} _ {[2]} +2 a^2 \\pi\r\n\\end{align}\\]\r\n\u3053\u3053\u3067\u4e0b\u7dda\u90e8 [2] \u306f, \u4e0b\u56f3\u659c\u7dda\u90e8\u306e\u9762\u7a4d\u3092\u8868\u3059\u306e\u3067<\/p>\r\n\"wsr20150502\"\r\n

    \\[\\begin{align}\r\n[2] & = \\dfrac{1}{2} \\cdot ( 2 \\sqrt{2} a )^2 \\cdot \\dfrac{\\pi}{6} +\\dfrac{1}{2} \\cdot \\sqrt{2} a \\cdot \\sqrt{6} a \\\\\r\n& = \\left( \\dfrac{2 \\pi}{3} +\\sqrt{3} \\right) a^2\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\nS & = 2 \\pi \\left( \\dfrac{2 \\pi}{3} +\\sqrt{3} \\right) a^2 +2 a^2 \\pi \\\\\r\n& = \\underline{2 \\pi \\left( \\dfrac{2 \\pi}{3} +\\sqrt{3} +1 \\right) a^2}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a \\gt 0\\) \u3068\u3059\u308b. \\(xy\\) \u5e73\u9762\u4e0a\u306b\u70b9 A \\(( -\\sqrt{2} a , 0 )\\) , B \\(( \\sqrt{2} a , 0 )\\) \u3092\u56fa\u5b9a\u3059\u308b. \u52d5\u70b9 P \\(( x , y )\\) \u306f … \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":[134],"tags":[147,137],"class_list":["post-1330","post","type-post","status-publish","format-standard","hentry","category-waseda_r_2015","tag-waseda_r","tag-137"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1330","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=1330"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1330\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1330"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1330"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1330"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}