{"id":1935,"date":"2021-10-19T15:48:54","date_gmt":"2021-10-19T06:48:54","guid":{"rendered":"https:\/\/www.roundown.net\/nyushi\/?p=1935"},"modified":"2021-10-19T15:49:35","modified_gmt":"2021-10-19T06:49:35","slug":"wsr201605","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/wsr201605\/","title":{"rendered":"\u65e9\u7a32\u7530\u7406\u5de52016\uff1a\u7b2c5\u554f"},"content":{"rendered":"<hr \/>\n<p>\\(xyz\\) \u7a7a\u9593\u4e0a\u306b\u70b9 A \\(( 0 , 0 , \\sqrt{3} )\\) \u3092\u3068\u308b.\r\n\\(xy\\) \u5e73\u9762\u4e0a\u306e\u70b9 P \\(( a , b , 0 )\\) \u306f, \u7dda\u5206 AP \u306e\u9577\u3055\u304c \\(2\\) \u3067, \\(a \\geqq 0\\) , \\(b \\geqq 0\\) \u3068\u306a\u308b\u3088\u3046\u306b\u52d5\u304f. \u3053\u306e\u3068\u304d\u7dda\u5206 AP \u304c\u3048\u304c\u304f\u56f3\u5f62\u3092 \\(F\\) \u3068\u3059\u308b. \u6b21\u306e\u554f\u306b\u7b54\u3048\u3088.<\/p>\r\n<ol>\r\n<li><p><strong>(1)<\/strong>\u3000\u70b9 P \u306e\u8ecc\u8de1\u3092 \\(xy\\) \u5e73\u9762\u4e0a\u306b\u56f3\u793a\u305b\u3088.<\/p><\/li>\r\n<li><p><strong>(2)<\/strong>\u3000\u70b9 Q \\(( x , y , z )\\) \u3092\u56f3\u5f62 \\(F\\) \u4e0a\u306e\u70b9\u3068\u3059\u308b\u3068\u304d, \\(z\\) \u3092 \\(x , y\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n<li><p><strong>(3)<\/strong>\u3000\u56f3\u5f62 \\(F\\) , \u5ea7\u6a19\u5e73\u9762 \\(x=0\\) , \\(y=0\\) , \\(z=0\\) \u306b\u3088\u3063\u3066\u56f2\u307e\u308c\u308b\u90e8\u5206\u3092 \\(x\\) \u8ef8\u306e\u5468\u308a\u306b \\(1\\) \u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u56de\u8ee2\u4f53\u3092 \\(V\\) \u3068\u3059\u308b. \\(V\\) \u306e\u5e73\u9762 \\(x=t\\) \u306b\u3088\u308b\u5207\u308a\u53e3\u306e\u9762\u7a4d \\(S(t)\\) \u3092 \\(t\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n<li><p><strong>(4)<\/strong>\u3000\\(V\\) \u306e\u4f53\u7a4d\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>\u25b3OAP \u306b\u7740\u76ee\u3059\u308c\u3070\r\n\\[\r\n\\text{OP} = \\sqrt{2^2 -3} = 1\r\n\\]\r\n\u306a\u306e\u3067, P \u306f O \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(1\\) \u306e\u5186\u5468\u306e\u7b2c \\(1\\) \u8c61\u9650\u90e8\u5206\u306b\u3042\u308a, \u8ecc\u8de1\u306f\u4e0b\u56f3.<\/p>\r\n<img decoding=\"async\" src=\"\/nyushi\/wp-content\/uploads\/wsr20160501.svg\" alt=\"wsr20160501\" class=\"aligncenter size-full\" \/>\r\n<p><strong>(2)<\/strong><\/p>\r\n<p>R \\(( 0 , 0 , z )\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\text{RQ} = \\sqrt{x^2 +y^2}\r\n\\]\r\n\\(\\triangle \\text{OAP} \\sim \\triangle \\text{RAQ}\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n( \\sqrt{3} -z ) : \\sqrt{x^2 +y^2} & = \\sqrt{3} : 1 \\\\\r\n\\sqrt{3} \\sqrt{x^2 +y^2} & = \\sqrt{3} -z \\quad ... [1]\\\\\r\n\\text{\u2234} \\quad z & = \\underline{\\sqrt{3} \\left( 1 -\\sqrt{x^2 +y^2} \\right)}\r\n\\end{align}\\]\r\n<p><strong>(3)<\/strong><\/p>\r\n<p>\\(t\\) \u306e\u3068\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u306f, \\(0 \\leqq t \\leqq 1\\) ... [2] .<br \/>\r\n\\(x=t\\) \u3092 [1] \u306b\u4ee3\u5165\u3057\u3066, \u4e21\u8fba\u3092\u5e73\u65b9\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n3 \\left( y^2 +t^2 \\right) & = \\left( \\sqrt{3} -z \\right)^2 \\\\\r\n\\text{\u2234} \\quad \\dfrac{\\left( \\sqrt{3} -z \\right)^2}{3 t^2} -\\dfrac{y^2}{t^2} & = 1\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u5e73\u9762 \\(x=t\\) \u306b\u304a\u3051\u308b\u56f3\u5f62 \\(F\\) \u306e\u65ad\u9762\u306f, \u53cc\u66f2\u7dda\u306e\u4e00\u90e8\u3068\u306a\u308b.<br \/>\r\n\u3053\u306e\u3068\u304d, \\(z\\) \u306e\u3068\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u306f\r\n\\[\r\n0 \\leqq z \\leqq \\sqrt{3} (1-t) \\quad ... [3]\r\n\\]\r\n\u3053\u306e\u66f2\u7dda\u4e0a\u306e\u70b9\u3068 \\(x\\) \u8ef8\u3068\u306e\u8ddd\u96e2\u306e \\(2\\) \u4e57\u3092 \\(f(z)\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nf(z) & = y^2 +z^2 \\\\\r\n& = \\dfrac{\\left( \\sqrt{3} -z \\right)^2}{3 t^2}{3} -t^2 +z^2 \\\\\r\n& = \\dfrac{4}{3} z^2 -\\dfrac{2 \\sqrt{3}}{3} z +1 -t^2 \\\\\r\n& = \\dfrac{4}{3} \\left( z -\\dfrac{\\sqrt{3}}{4} \\right)^2 +\\dfrac{3}{4} -t^2\r\n\\end{align}\\]\r\n[3] \u306b\u304a\u3051\u308b \\(f(z)\\) \u306e\u6700\u5927\u5024\u306e\u5019\u88dc\u306f\r\n\\[\\begin{align}\r\nf(0) & = 1 -t^2 \\ , \\\\\r\nf \\left( \\sqrt{3} (1-t) \\right) & = 4 ( 1 -t )^2 -2 ( 1 -t ) +1 -t^2 \\\\\r\n& = 3 ( 1 -t )^2\r\n\\end{align}\\]\r\n[2] \u306e\u7bc4\u56f2\u3067, \u3053\u308c\u3089\u306e\u5927\u5c0f\u3092\u6bd4\u8f03\u3059\u308b\u3068<\/p>\r\n<ul>\r\n<li><p>\\(0 \\leqq t \\lt \\dfrac{1}{2}\\) \u306e\u3068\u304d, \\(f \\left( \\sqrt{3} (1-t) \\right)\\) \u304c\u6700\u5927<\/p><\/li>\r\n<li><p>\\(\\dfrac{1}{2} \\leqq t \\leqq 1\\) \u306e\u3068\u304d, \\(f(0)\\) \u304c\u6700\u5927<\/p><\/li>\r\n<\/ul>\r\n<p>\u3088\u3063\u3066, \u6c42\u3081\u308b\u9762\u7a4d\u306f\r\n\\[\r\nS(t) = \\underline{\\left\\{ \\begin{array}{ll} 3 \\pi ( 1 -t )^2 & \\left( \\ 0 \\leqq t \\lt \\dfrac{1}{2} \\ \\text{\u306e\u3068\u304d} \\ \\right) \\\\ \\pi ( 1 -t ^2 ) & \\left( \\ \\dfrac{1}{2} \\leqq t \\leqq 1 \\ \\text{\u306e\u3068\u304d} \\ \\right) \\end{array} \\right.}\r\n\\]\r\n<p><strong>(4)<\/strong><\/p>\r\n<p><strong>(3)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nV & = \\displaystyle\\int _ {0}^{1} S(t) \\, dt \\\\\r\n& = 3 \\pi \\displaystyle\\int _ {0}^{\\frac{1}{2}} ( 1 -t )^2 \\, dt +\\pi \\displaystyle\\int _ {\\frac{1}{2}}^{1} ( 1 -t ^2 ) \\, dt \\\\\r\n& = \\pi \\left[ ( t -1 )^3 \\right] _ {0}^{\\frac{1}{2}} +\\pi \\left[ t -\\dfrac{t^3}{3} \\right] _ {\\frac{1}{2}}^{1} \\\\\r\n& = \\pi \\left( 1 -\\dfrac{1}{8} \\right) +\\pi \\left( \\dfrac{2}{3} -\\dfrac{1}{2} +\\dfrac{1}{24} \\right) \\\\\r\n& = \\underline{\\dfrac{13}{12} \\pi}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xyz\\) \u7a7a\u9593\u4e0a\u306b\u70b9 A \\(( 0 , 0 , \\sqrt{3} )\\) \u3092\u3068\u308b. \\(xy\\) \u5e73\u9762\u4e0a\u306e\u70b9 P \\(( a , b , 0 )\\) \u306f, \u7dda\u5206 AP \u306e\u9577\u3055\u304c \\(2\\) \u3067, \\(a \\geq &hellip; <a href=\"https:\/\/www.roundown.net\/nyushi\/wsr201605\/\">\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":[157],"tags":[147,162],"class_list":["post-1935","post","type-post","status-publish","format-standard","hentry","category-waseda_r_2016","tag-waseda_r","tag-162"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1935","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=1935"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1935\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1935"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1935"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1935"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}