(1)<\/strong>\u3000\\(\\dfrac{dx}{dt}\\) \u304a\u3088\u3073 \\(\\dfrac{dy}{dt}\\) \u3092\u8a08\u7b97\u3057, \\(C\\) \u306e\u6982\u5f62\u3092\u56f3\u793a\u305b\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(C\\) \u304c \\(x\\) \u8ef8\u3068 \\(y\\) \u8ef8\u3067\u56f2\u307e\u308c\u305f\u90e8\u5206\u306e\u9762\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\[\\begin{align}\r\n\\dfrac{dx}{dt} & = -3 \\sin t +3 \\sin 3t \\\\\r\n& = -3 \\sin t +3 ( 3 \\sin t -4 \\sin^3 t ) \\\\\r\n& = 6 \\sin t ( 1 -2 \\sin^2 t ) \\\\\r\n& = \\sin t \\left( \\dfrac{1}{\\sqrt{2}} +\\sin t \\right) \\left( \\dfrac{1}{\\sqrt{2}} -\\sin t \\right)\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\n\\dfrac{dy}{dt} & = 3 \\cos t -3 \\cos 3t \\\\\r\n& = 3 \\cos t +3 ( 4 \\cos^3 t -3 \\cos t ) \\\\\r\n& = 12 \\cos t ( 1 -\\cos^2 t ) \\\\\r\n& = 12 \\sin t ( 1 +\\cos t ) ( 1 -\\cos t )\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(0 \\leqq t \\leqq \\dfrac{\\pi}{2}\\) \u306b\u304a\u3051\u308b \\(x,y\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} t & 0 & \\cdots & \\dfrac{\\pi}{4} & \\cdots & \\dfrac{\\pi}{2} \\\\ \\hline \\dfrac{dx}{dt} & 0 & + & 0 & - & \\\\ \\hline x & 2 & \\rightarrow & 2 \\sqrt{2} & \\leftarrow & 0 \\\\ \\hline \\dfrac{dy}{dt} & 0 & + & & + & 0 \\\\ \\hline y & 0 & \\uparrow & \\sqrt{2} & \\uparrow & 4 \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \\(C\\) \u306e\u6982\u5f62\u306f\u4e0b\u56f3.<\/p>\r\n (2)<\/strong><\/p>\r\n \u6c42\u3081\u308b\u9762\u7a4d \\(S\\) \u306f\r\n\\[\\begin{align}\r\nS & = \\displaystyle\\int _ 0^4 x \\, dy \\\\\r\n& = \\displaystyle\\int _ {0}^{\\frac{\\pi}{2}} ( 3 \\cos t -\\cos 3t ) ( 3 \\cos t -3 \\cos 3t ) \\, dt \\\\\r\n& = 3 \\displaystyle\\int _ {0}^{\\frac{\\pi}{2}} ( 3 \\cos^2 t +\\cos^2 3t -4 \\cos t \\cos 3t ) \\, dt \\\\\r\n& = 3 \\displaystyle\\int _ {0}^{\\frac{\\pi}{2}} \\left\\{ \\dfrac{3 ( \\cos 2t +1 )}{2} +\\dfrac{\\cos 6t +1}{2} -2 ( \\cos 4t +\\cos 2t ) \\right\\} \\, dt \\\\\r\n& = 3 \\left[ 2t +\\dfrac{1}{12} \\sin 6t -\\dfrac{1}{2} \\sin 4t -\\dfrac{1}{4} \\sin 2t \\right] _ {0}^{\\frac{\\pi}{2}} \\\\\r\n& = \\underline{3 \\pi}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u6b21\u306e\u3088\u3046\u306b\u5a92\u4ecb\u5909\u6570\u8868\u793a\u3055\u308c\u305f \\(xy\\) \u5e73\u9762\u4e0a\u306e\u66f2\u7dda\u3092 \\(C\\) \u3068\u3059\u308b: \\[ \\left\\{ \\begin{array}{l} x = 3 \\cos t -\\cos 3t \\\\ y = 3 \\sin t -\\s … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
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