{"id":180,"date":"2011-12-02T22:42:08","date_gmt":"2011-12-02T13:42:08","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=180"},"modified":"2021-03-24T08:20:10","modified_gmt":"2021-03-23T23:20:10","slug":"kyr200905","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/kyr200905\/","title":{"rendered":"\u4eac\u5927\u7406\u7cfb\u4e592009\uff1a\u7b2c5\u554f"},"content":{"rendered":"
\n

\\(xy\\) \u5e73\u9762\u4e0a\u3067\u539f\u70b9\u3092\u6975, \\(x\\) \u8ef8\u306e\u6b63\u306e\u65b9\u5411\u3092\u59cb\u7dda\u3068\u3059\u308b\u6975\u5ea7\u6a19\u306b\u95a2\u3057\u3066,\r\n\u6975\u65b9\u7a0b\u5f0f \\(r = 2 +\\cos \\theta \\quad \\left( 0 \\leqq \\theta \\leqq \\pi \\right)\\) \u306b\u3088\u308a\u8868\u3055\u308c\u308b\u66f2\u7dda\u3092 \\(C\\) \u3068\u3059\u308b. \\(C\\) \u3068 \\(x\\) \u8ef8\u3068\u3067\u56f2\u307e\u308c\u305f\u56f3\u5f62\u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3057\u3066\u5f97\u3089\u308c\u308b\u7acb\u4f53\u306e\u4f53\u7a4d\u3092\u6c42\u3081\u3088.<\/p>\r\n

\r\n\r\n

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

\\(C\\) \u306e\u6982\u5f62\u306f\u4e0a\u56f3\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\\begin{align}\r\nx & = r \\cos \\theta =2 \\cos \\theta +\\cos^2 \\theta \\\\\r\ny & = r \\sin \\theta =2 \\sin \\theta +\\sin \\theta \\cos \\theta\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\r\n\\dfrac{dx}{d \\theta} = -2 \\sin \\theta -2 \\cos \\theta \\sin \\theta\r\n\\]\r\n\u3057\u305f\u304c\u3063\u3066, \u6c42\u3081\u308b\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\\begin{align}\r\nV & = \\pi \\displaystyle\\int _ {-1}^3 y^2 \\, dx \\\\\r\n& =\\pi \\displaystyle\\int _ {\\pi}^0 \\left( 2 \\sin \\theta +\\sin \\theta \\cos \\theta \\right)^2 \\left( -2 \\sin \\theta -2 \\cos \\theta \\sin \\theta \\right) d \\theta \\\\\r\n& =-2 \\pi \\displaystyle\\int _ \\pi^{0} \\sin^3 \\theta \\left( 2 +\\cos \\theta \\right)^2 \\left( 1 +\\cos \\theta \\right) d \\theta \\\\\r\n& =2 \\pi \\displaystyle\\int _ {\\pi}^{0} \\left( 1- \\cos^2 \\theta \\right) \\left( 2 +\\cos \\theta \\right)^2 \\left( 1 +\\cos \\theta \\right) \\left( - \\sin \\theta \\right) d \\theta\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \\(t= \\cos \\theta\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{gather}\r\ndt = -\\sin \\theta \\, d \\theta , \\\\\r\n\\begin{array}{c|ccc} \\theta & \\pi & \\rightarrow & 0 \\\\ \\hline t & -1 & \\rightarrow & 1 \\end{array}\r\n\\end{gather}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nV & = 2\\pi \\displaystyle\\int _ {-1}^1 (1-t^2)(2+t)^2(1+t) \\, dt \\\\\r\n& = 2\\pi \\displaystyle\\int _ {-1}^1 \\left( -t^5-5t^4-7t^3+t^2+8t+4 \\right) dt\\ \\\\\r\n& = 4\\pi \\displaystyle\\int _ 0^1 \\left( -5t^4+t^2+4 \\right) dt \\\\\r\n& = 4\\pi \\left[ -t^5 +\\dfrac{t^3}{3} +4t \\right] _ 0^1 \\\\\r\n& = \\underline{\\dfrac{40 \\pi}{3}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u4e0a\u3067\u539f\u70b9\u3092\u6975, \\(x\\) \u8ef8\u306e\u6b63\u306e\u65b9\u5411\u3092\u59cb\u7dda\u3068\u3059\u308b\u6975\u5ea7\u6a19\u306b\u95a2\u3057\u3066, \u6975\u65b9\u7a0b\u5f0f \\(r = 2 +\\cos \\theta \\quad \\left( 0 \\leqq \\theta \\leqq \\pi \\r … \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":[21],"tags":[140,15],"class_list":["post-180","post","type-post","status-publish","format-standard","hentry","category-kyoto_r_2009","tag-kyoto_r","tag-15"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/180","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=180"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/180\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=180"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=180"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=180"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}