\r\n \r\n
\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\\[\\begin{align}\r\n\\dfrac{dx}{dt} & = -2\\sin 2t , \\\\\r\n\\dfrac{dy}{dt} & = \\sin t +t \\cos t\r\n\\end{align}\\]\r\n\\(0 \\leqq t \\leqq 2 \\pi\\) \u306b\u304a\u3044\u3066\r\n\\[\\begin{align}\r\n\\dfrac{dx}{dt} & = -2\\sin 2t =0 \\\\\r\n\\text{\u2234} \\quad t =0 , & \\dfrac{\\pi}{2} , \\pi , \\dfrac{3 \\pi}{2} , 2 \\pi\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\n\\dfrac{dy}{dt} = \\sin t +t \\cos t & =0 \\\\\r\n\\text{\u2234} \\quad \\tan t & = -t \\quad ... [1]\r\n\\end{align}\\]\r\n\\(y= \\tan t\\) \u3068 \\(y =-t\\) \u306e\u30b0\u30e9\u30d5\u306f\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n![\"\"](\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/tokyo_r_2008_06_01.png\" "\\"tokyo_r_2008_06_01\\"")
\r\n
\u3057\u305f\u304c\u3063\u3066, [1] \u306e\u89e3\u306f\r\n\\[\r\nt = 0 , \\alpha , \\beta \\quad \\left( \\ \\dfrac{\\pi}{2} \\lt \\alpha \\lt \\pi , \\dfrac{3 \\pi}{2} \\lt \\beta <2 \\pi \\ \\right)\r\n\\]\r\n\u4ee5\u4e0a\u3088\u308a, \\((x, y)\\) \u306e\u5897\u6e1b\u8868\u306f\u4e0b\u8868\u306e\u901a\u308a\u3068\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccccccccccc} t & 0 & \\cdots & \\dfrac{\\pi}{2} & \\cdots & \\alpha & \\cdots & \\pi & \\cdots & \\dfrac{3 \\pi}{2} & \\cdots & \\beta & \\cdots & 2\\pi \\\\ \\hline \\frac{dx}{dt} & 0 & - & 0 & + & & + & 0 & - & 0 & + & & + & 0 \\\\ \\hline \\frac{dy}{dt} & 0 & + & & + & 0 & - & & - & & - & 0 & + & \\\\ \\hline x & 1 & \\leftarrow & -1 & \\rightarrow & & \\rightarrow & 1 & \\leftarrow & -1 & \\rightarrow & & \\rightarrow & 1 \\\\ \\hline y & 0 & \\uparrow & \\dfrac{\\pi}{2} & \\uparrow & \\max & \\downarrow & 0 & \\downarrow & -\\dfrac{3 \\pi}{2} & \\downarrow & \\min & \\uparrow & 0 \\end{array}\r\n\\]\r\n\u3086\u3048\u306b, \u4e0e\u3048\u3089\u308c\u305f\u66f2\u7dda\u306e\u6982\u5f62\u306f\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n![\"\"](\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/tokyo_r_2008_06_02.png\" "\\"tokyo_r_2008_06_02\\"")
\r\n
\u3057\u305f\u304c\u3063\u3066, \u6c42\u3081\u308b\u9762\u7a4d \\(S\\) \u306f\r\n\\[\\begin{align}\r\nS & = \\displaystyle\\int _ {t=\\frac{\\pi}{2}}^{t=\\pi} y \\, dx -\\displaystyle\\int _ {t=\\frac{\\pi}{2}}^{t=0} y \\, dx +\\displaystyle\\int _ {t=\\frac{3 \\pi}{2}}^{t= 2\\pi} (-y) \\, dx -\\displaystyle\\int _ {t=\\frac{3 \\pi}{2}}^{t=\\pi} (-y) \\, dx \\\\\r\n& = \\displaystyle\\int _ {t=0}^{t=\\pi} y \\, dx +\\displaystyle\\int _ {t=2 \\pi}^{t=\\pi} y \\, dx\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \\(F(t) =\\displaystyle\\int y \\, dx\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nF(t) & =\\displaystyle\\int \\, t \\sin t ( -2 \\sin 2t ) \\, dt \\\\\r\n& = -4 \\displaystyle\\int \\, t \\sin^2 t ( \\sin t )' \\, dt \\\\\r\n& = -\\dfrac{4}{3} t \\sin^3 t +\\dfrac{4}{3} \\displaystyle\\int \\, \\sin^3 t \\, dt \\\\\r\n& = -\\dfrac{4}{3} t \\sin^3 t -\\dfrac{4}{3} \\displaystyle\\int \\, \\left( 1 -\\cos^2 t \\right) \\left( \\cos t \\right)' \\, dt \\\\\r\n& = -\\dfrac{4}{3} t \\sin^3 t -\\dfrac{4}{3} \\left( \\cos t -\\dfrac{1}{3} \\cos^3 t \\right) \\\\\r\n& = -\\dfrac{4}{3} t \\sin^3 t -\\dfrac{4}{3} \\cos t +\\dfrac{4}{9} \\cos^3 t +C \\quad ( \\ C \\text{\u306f\u7a4d\u5206\u5b9a\u6570} )\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\nS & = 2F( \\pi ) -F(0) -F( 2\\pi ) \\\\\r\n& = 2 \\left( \\dfrac{4}{3} -\\dfrac{4}{9} \\right) -\\left( -\\dfrac{4}{3} +\\dfrac{4}{9} \\right) -\\left( -\\dfrac{4}{3} +\\dfrac{4}{9} \\right) \\\\\r\n& = \\underline{\\dfrac{32}{9}}\r\n\\end{align}\\]\r\n\r\n
\r\n « \u89e3\u7b54\u3092\u96a0\u3059 <\/a>\r\n <\/p>\r\n <\/div>","protected":false},"excerpt":{"rendered":"\u5ea7\u6a19\u5e73\u9762\u306b\u304a\u3044\u3066, \u5a92\u4ecb\u5909\u6570 \\(t\\) \u3092\u7528\u3044\u3066 \\[ \\left\\{ \\begin{array}{l} x= \\cos 2t \\\\ y= t \\sin t \\end{array} \\right. \\quad ( \\ […]","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":[17],"tags":[139,16],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/255"}],"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=255"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/255\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=255"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=255"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=255"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}