{"id":38,"date":"2011-11-25T22:04:42","date_gmt":"2011-11-25T13:04:42","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=38"},"modified":"2021-09-24T18:01:42","modified_gmt":"2021-09-24T09:01:42","slug":"tok201104","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tok201104\/","title":{"rendered":"\u6771\u5de5\u59272011\uff1a\u7b2c4\u554f"},"content":{"rendered":"<hr \/>\n<p>\u5e73\u9762\u4e0a\u306b\u4e00\u8fba\u306e\u9577\u3055\u304c \\(1\\) \u306e\u6b63\u65b9\u5f62 \\(D\\) \u304a\u3088\u3073 \\(D\\) \u3068\u4ea4\u308f\u308b\u76f4\u7dda\u304c\u3042\u308b\u3068\u3059\u308b.\r\n\u3053\u306e\u76f4\u7dda\u3092\u8ef8\u306b \\(D\\) \u3092\u56de\u8ee2\u3057\u3066\u5f97\u3089\u308c\u308b\u56de\u8ee2\u4f53\u306b\u3064\u3044\u3066\u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\\(D\\) \u3068\u540c\u3058\u5e73\u9762\u4e0a\u306e\u76f4\u7dda \\(l\\) \u306f \\(D\\) \u306e\u3069\u306e\u8fba\u306b\u3082\u5e73\u884c\u3067\u306a\u3044\u3082\u306e\u3068\u3059\u308b.\r\n\u8ef8\u3068\u3059\u308b\u76f4\u7dda\u306f \\(l\\) \u3068\u5e73\u884c\u306a\u3082\u306e\u306e\u4e2d\u3067\u8003\u3048\u308b\u3068\u304d, \u56de\u8ee2\u4f53\u306e\u4f53\u7a4d\u3092\u6700\u5927\u306b\u3059\u308b\u76f4\u7dda\u306f \\(D\\) \u3068\u552f \\(1\\) \u70b9\u3067\u4ea4\u308f\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\\(D\\) \u3068\u4ea4\u308f\u308b\u76f4\u7dda\u3092\u8ef8\u3068\u3057\u3066\u3067\u304d\u308b\u3059\u3079\u3066\u306e\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d\u306e\u4e2d\u3067\u6700\u5927\u3068\u306a\u308b\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n<hr>\r\n<!--more-->\r\n<h4>\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n<p><strong>(1)<\/strong><\/p>\r\n<p>\u56f3\u5f62 \\(R\\) \u306f\u9818\u57df \\(0 \\leqq x \\leqq a\\) \u306b\u3042\u308a, \u76f4\u7dda \\(x = k\\) \u306b\u542b\u307e\u308c\u308b\u90e8\u5206\u306e\u9577\u3055\uff08\u4ee5\u5f8c, \u5e45\u3068\u3088\u3076\uff09\u304c \\(f(k)\\) \u3067\u3042\u308b\u3068\u3059\u308b.<br \/>\r\n\u3053\u306e\u3068\u304d, \u56f3\u5f62 \\(R\\) \u306e \\(y\\) \u8ef8\u3092\u4e2d\u5fc3\u3068\u3057\u305f\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\r\nV = 2\\pi \\displaystyle\\int _ 0^a x f(x) dx\r\n\\]\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/toko_2011_04_01.png\" alt=\"\" title=\"toko_2011_04_01\" class=\"aligncenter size-full\" \/>\r\n<p>\u4ee5\u4e0b\u3067\u306f \\(2\\) \u3064\u306e\u88dc\u984c [A], [B] \u3092\u793a\u3059.<\/p>\r\n<ul>\r\n<li><p>\u88dc\u984c [A] \uff1a<br \/>\r\n\\(2\\) \u3064\u306e\u56f3\u5f62 \\(R _ 1 , R _ 2\\) \u306f\u305d\u308c\u305e\u308c\u9818\u57df \\(0 \\leqq x \\leqq a , \\ b \\leqq x \\leqq a+b\\) \u306b\u3042\u308a, \u5e45\u304c\u305d\u308c\u305e\u308c \\(f(x) , g(x)\\) \u3067\u3042\u308b\u3068\u3059\u308b.<br \/>\r\n\u3053\u306e\u3068\u304d, \\(f(x+b) = g(x)\\) \u3067\u3042\u308c\u3070, \u56f3\u5f62 \\(R _ 1 , R _ 2\\) \u306e \\(y\\) \u8ef8\u3092\u4e2d\u5fc3\u306b\u3057\u305f\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d\u3092\u305d\u308c\u305e\u308c \\(V _ 1 , V _ 2\\) \u3068\u3059\u308c\u3070\r\n\\[\r\nV _ 1 \\lt V _ 2\r\n\\]\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/toko_2011_04_02.png\" alt=\"\" title=\"toko_2011_04_02\" class=\"aligncenter size-full\" \/>\r\n<ol>\r\n<li>[A] \u306e\u8a3c\u660e\uff1a<br \/>\r\n\\[\\begin{align}\r\nV _ 1 & = 2\\pi \\displaystyle\\int _ 0^a x f(x) dx \\\\\r\nV _ 2 & = 2\\pi \\displaystyle\\int _ b^{a+b} x f(x+b) dx\r\n\\end{align}\\]\r\n\u3053\u3053\u3067 \\(u = x-b\\) \u3068\u304a\u3051\u3070\r\n\\[\\begin{gather}\r\ndx = du , \\\\\r\n\\begin{array}{c|ccc} x & b & \\rightarrow & a+b \\\\ \\hline u & 0 & \\rightarrow & a \\end{array}\r\n\\end{gather}\\]\r\n\u3086\u3048\u306b\r\n\\[\\begin{align}\r\nV _ 2 & = 2\\pi \\displaystyle\\int _ 0^a (u+b) f(u) \\, du \\\\\r\n& = V _ 1 +2b\\pi \\displaystyle\\int _ 0^a f(u) \\, du \\gt V _ 1\r\n\\end{align}\\]<\/li>\r\n<\/ol><\/p><\/li>\r\n<li><p>\u88dc\u984c [B] \uff1a<br \/>\r\n\u56f3\u5f62 \\(R\\) \u306f\u9818\u57df \\(0 \\leqq x \\leqq a\\) \u306b\u3042\u308a, \u5e45 \\(f(x)\\) \u304c\u5358\u8abf\u5897\u52a0\u3067\u3042\u308b\u3068\u3059\u308b.<br \/>\r\n\u3053\u306e\u3068\u304d, \u56f3\u5f62\u9818\u57df \\(R\\) \u306e \\(y\\) \u8ef8, \u76f4\u7dda \\(x = a\\) \u3092\u4e2d\u5fc3\u306b\u3057\u305f\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d\u3092\u305d\u308c\u305e\u308c \\(V _ 1 , V _ 2\\) \u3068\u3059\u308c\u3070\r\n\\[\r\nV _ 1 \\gt V _ 2\r\n\\]\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/toko_2011_04_03.png\" alt=\"\" title=\"toko_2011_04_03\" class=\"aligncenter size-full\" \/>\r\n<ol>\r\n<li>[B] \u306e\u8a3c\u660e\uff1a\r\n\\[\\begin{align}\r\nV _ 1 & = 2\\pi \\displaystyle\\int _ 0^a x f(x) \\, dx , \\\\\r\nV _ 2 & = 2\\pi \\displaystyle\\int _ 0^a x f(a-x) \\, dx\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\n\\dfrac{V _ 1 -V _ 2}{2\\pi} & = \\displaystyle\\int _ 0^a x \\left\\{ f(x) -f(a-x) \\right\\} \\, dx \\\\\r\n& = \\displaystyle\\int _ 0^{\\frac{a}{2}} x \\left\\{ f(x) -f(a-x) \\right\\} \\, dx \\\\\r\n& \\qquad +\\underline{\\displaystyle\\int _ {\\frac{a}{2}}^a x \\left\\{ f(x) -f(a-x) \\right\\} \\, dx} _ {[1]}\r\n\\end{align}\\]\r\n\u4e0b\u7dda\u90e8 [1] \u306b\u3064\u3044\u3066, \\(u = a-x\\) \u3068\u304a\u3051\u3070\r\n\\[\\begin{gather}\r\ndx = -du , \\\\\r\n\\begin{array}{c|ccc} x & \\dfrac{a}{2} & \\rightarrow & a \\\\ \\hline u & \\dfrac{a}{2} & \\rightarrow & 0 \\end{array}\r\n\\end{gather}\\]\r\n\u3086\u3048\u306b\r\n\\[\\begin{align}\r\n[1] & = \\displaystyle\\int _ 0^{\\frac{a}{2}} (a-u) \\left\\{ f(a-u) -f(u) \\right\\} \\, du \\\\\r\n& = \\displaystyle\\int _ 0^{\\frac{a}{2}} (u-a) \\left\\{ f(u) -f(a-u) \\right\\} \\, du\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n\\dfrac{V _ 1 -V _ 2}{2\\pi} = \\displaystyle\\int _ 0^{\\frac{a}{2}} (a-2x) \\left\\{ f(a-x) -f(x) \\right\\} \\, dx\r\n\\]\r\n\u3053\u3053\u3067, \\(0 \\leqq x \\leqq \\dfrac{a}{2}\\) \u306e\u3068\u304d, \\(2x-a \\geqq 0 \\quad ... [1]\\) .<br \/>\r\n\u307e\u305f \\(f(x)\\) \u306f\u5358\u8abf\u5897\u52a0\u306a\u306e\u3067, \\(f(a-x) -f(x) \\geqq 0 \\quad ... [2]\\) .<br \/>\r\n\u5e38\u306b [1] [2] \u3068\u3082\u306b\u7b49\u53f7\u6210\u7acb\u3059\u308b\u3053\u3068\u306f\u306a\u3044\u306e\u3067\r\n\\[\\begin{align}\r\n\\dfrac{V _ 1 -V _ 2}{2\\pi} \\gt 0 \\\\\r\n\\text{\u2234} \\quad V _ 1 \\gt V _ 2\r\n\\end{align}\\]<\/li>\r\n<\/ol><\/p><\/li>\r\n<\/ul>\r\n<p>\u3053\u3053\u304b\u3089, \u984c\u610f\u3092\u793a\u3059.<br \/>\r\n\u5bfe\u79f0\u6027\u304b\u3089, \u8ef8\u306e\u4f4d\u7f6e\u306f \\(D\\) \u306e\u4e2d\u5fc3 O \u3092\u901a\u308b\u76f4\u7dda\u3088\u308a\u5de6\u5074\u306e\u307f\u3092\u8003\u3048\u308c\u3070\u3088\u3044.<br \/>\r\n\u3082\u3063\u3068\u3082\u5de6\u306e \\(1\\) \u9802\u70b9\u3092\u901a\u308b\u76f4\u7dda\u3092 \\(L\\) , \u305d\u306e\u4ed6\u306e\u76f4\u7dda\u3092 \\(K\\) \u3068\u304a\u304d, \u305d\u308c\u305e\u308c\u3092\u4e2d\u5fc3\u306b\u3057\u305f \\(D\\) \u306e\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d\u3092 \\(V _ L , V _ K\\) \u3068\u304a\u304f.<br \/>\r\n\\(D\\) \u304c \\(K\\) \u306b\u3088\u3063\u3066\u5206\u3051\u3089\u308c\u308b\u90e8\u5206\u306e\u5de6\u5074\u3092 \\(A\\) , \u53f3\u5074\u3092 \\(B\\) \u3068\u304a\u304d, \u5404\u90e8\u5206\u306e, \\(L , K\\) \u3092\u4e2d\u5fc3\u306b\u3057\u305f\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d\u3092\u305d\u308c\u305e\u308c \\(V _ {LA}\\) \u3068 \\(V _ {LB}\\) , \\(V _ {KA}\\) \u3068 \\(V _ {KB}\\) \u3068\u304a\u304f.<\/p>\r\n<img decoding=\"async\" src=\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/toko_2011_04_04.png\" alt=\"\" title=\"toko_2011_04_04\" class=\"aligncenter size-full\" \/>\r\n<p>\u3053\u306e\u3068\u304d\r\n\\[\\begin{align}\r\nV _ L & = V _ {LA} +V _ {LB} \\\\\r\nV _ K & \\lt V _ {KA} +V _ {KB} \\quad ( \\ \\text{\u2235} \\ \\text{\u56de\u8ee2\u4f53\u3069\u3046\u3057\u306b\u91cd\u306a\u308a\u304c\u3042\u308b} )\r\n\\end{align}\\]\r\n\u88dc\u984c [A] \u3088\u308a\r\n\\[\r\nV _ {LB} \\gt V _ {KB}\r\n\\]\r\n\u90e8\u5206 \\(A\\) \u306f\u53f3\u306b\u3044\u304f\u307b\u3069\u8ef8\u65b9\u5411\u306e\u5e45\u304c\u5358\u8abf\u5897\u52a0\u3057\u3066\u3044\u308b\u306e\u3067, \u88dc\u984c [B] \u3088\u308a\r\n\\[\r\nV _ {LA} \\gt V _ {KA}\r\n\\]\r\n\u4ee5\u4e0a\u3088\u308a\r\n\\[\r\nV _ L = V _ {LA} +V _ {LB} \\gt V _ {KA} +V _ {KB} \\gt V _ K\r\n\\]\r\n\u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n<p><strong>(2)<\/strong><\/p>\r\n","protected":false},"excerpt":{"rendered":"\u5e73\u9762\u4e0a\u306b\u4e00\u8fba\u306e\u9577\u3055\u304c \\(1\\) \u306e\u6b63\u65b9\u5f62 \\(D\\) \u304a\u3088\u3073 \\(D\\) \u3068\u4ea4\u308f\u308b\u76f4\u7dda\u304c\u3042\u308b\u3068\u3059\u308b. \u3053\u306e\u76f4\u7dda\u3092\u8ef8\u306b \\(D\\) \u3092\u56de\u8ee2\u3057\u3066\u5f97\u3089\u308c\u308b\u56de\u8ee2\u4f53\u306b\u3064\u3044\u3066\u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048\u3088. (1)\u3000\\(D\\) \u3068\u540c\u3058\u5e73\u9762\u4e0a\u306e\u76f4 &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/tok201104\/\">\u7d9a\u304d\u3092\u8aad\u3080 <span class=\"meta-nav\">&rarr;<\/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":[26],"tags":[141,13],"class_list":["post-38","post","type-post","status-publish","format-standard","hentry","category-toko_2011","tag-toko","tag-13"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/38","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=38"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/38\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=38"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=38"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=38"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}