\\(xyz\\) \u7a7a\u9593\u5185\u306e\u70b9 P \\(( 1, 0, 1 )\\) \u3068, \\(xy\\) \u5e73\u9762\u4e0a\u306e\u5186 \\(C : \\ x^2 +(y-2)^2 = 1\\) \u306b\u5c5e\u3059\u308b\u70b9 Q \\(( \\cos \\theta , 2 +\\sin \\theta , 0 )\\) \u3092\u8003\u3048\u308b.<\/p>\r\n
(1)<\/strong>\u3000\u76f4\u7dda PQ \u3068\u5e73\u9762 \\(z = t\\) \u306e\u4ea4\u70b9\u306e\u5ea7\u6a19\u3092 \\(( \\alpha , \\beta , t )\\) \u3068\u3059\u308b\u3068\u304d, \\(\\alpha^2 +\\beta^2\\) \u3092 \\(t\\) \u3068 \\(\\theta\\) \u3067\u8868\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\u7dda\u5206 PQ \u3092 \\(z\\) \u8ef8\u306e\u5468\u308a\u306b \\(1\\) \u56de\u8ee2\u3055\u305b\u3066\u3067\u304d\u308b\u66f2\u9762\u3068\u5e73\u9762 \\(z = 0\\) , \\(z = 1\\) \u306b\u3088\u3063\u3066\u56f2\u307e\u308c\u308b\u7acb\u4f53\u306e\u4f53\u7a4d\u3092 \\(\\theta\\) \u3067\u8868\u305b.<\/p><\/li>\r\n (3)<\/strong>\u3000Q \u304c \\(C\\) \u4e0a\u3092\u4e00\u5468\u3059\u308b\u3068\u304d, (2)<\/strong> \u3067\u6c42\u3081\u305f\u4f53\u7a4d\u306e\u6700\u5927\u5024, \u6700\u5c0f\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u70b9 \\(( \\alpha , \\beta , t )\\) \u306f\u7dda\u5206 QP \u3092 \\(t : (1-t)\\) \u306b\u5185\u5206\u3059\u308b\u306e\u3067\r\n\\[\\begin{align}\r\n\\alpha & = t \\cdot 1 +(1-t) \\cos \\theta = t +(1-t) \\cos \\theta , \\\\\r\n\\beta & = t \\cdot 0 +(1-t)( 2 +\\sin \\theta ) = (1-t)( 2 +\\sin \\theta )\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\\begin{align}\r\n\\alpha^2 & +\\beta^2 \\\\\r\n& = t^2 +2t(1-t) \\cos \\theta +(1-t)^2 \\cos^2 \\theta +(1-t)^2(2+\\sin \\theta)^2 \\\\\r\n& = \\underline{t^2 +2t(1-t) \\cos \\theta +(1-t)^2 ( 5 +4\\sin \\theta )}\r\n\\end{align}\\]\r\n (2)<\/strong><\/p>\r\n \u6c42\u3081\u308b\u4f53\u7a4d\u3092 \\(V\\) \u3068\u304a\u3051\u3070\r\n\\[\\begin{align}\r\nV & = \\pi \\displaystyle\\int _ 0^1 \\left( \\alpha^2 +\\beta^2 \\right) \\, dt \\\\\r\n& = \\pi \\displaystyle\\int _ 0^1 t^2 \\, dt +2 \\pi \\cos \\theta \\displaystyle\\int _ 0^1 t(1-t) \\, dt +\\pi ( 5 +4 \\sin \\theta ) \\displaystyle\\int _ 0^1 t^2 \\, dt \\\\\r\n& =\\dfrac{\\pi}{3} +\\dfrac{\\pi \\cos \\theta}{3} +\\dfrac{\\pi ( 5 +4\\sin \\theta )}{3} \\\\\r\n& =\\underline{\\dfrac{6 +4\\sin \\theta +\\cos \\theta}{3} \\pi}\r\n\\end{align}\\]\r\n (3)<\/strong><\/p>\r\n \\[\\begin{gather}\r\n4 \\sin \\theta +\\cos \\theta = \\sqrt{17} \\sin \\left( \\theta +\\alpha \\right) \\\\\r\n\\left( \\ \\text{\u305f\u3060\u3057} \\ \\sin \\alpha = \\dfrac{1}{\\sqrt{17}} , \\ \\sin \\alpha = \\dfrac{4}{\\sqrt{17}} \\right)\r\n\\end{gather}\\]\r\n\\(0 \\leqq \\theta \\lt 2 \\pi\\) \u3068\u52d5\u304f\u3068\u304d \\(-1 \\leqq \\sin \\left( \\theta +\\alpha \\right) \\leqq 1\\) \u306a\u306e\u3067, (2)<\/strong> \u306e\u7d50\u679c\u3068\u3042\u308f\u305b\u3066, \\(V\\) \u306b\u3064\u3044\u3066\r\n\\[\r\n\\text{\u6700\u5927\u5024} : \\ \\underline{\\dfrac{6 +\\sqrt{17}}{6} \\pi} , \\ \\text{\u6700\u5c0f\u5024} : \\ \\underline{\\dfrac{6 -\\sqrt{17}}{6} \\pi}\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xyz\\) \u7a7a\u9593\u5185\u306e\u70b9 P \\(( 1, 0, 1 )\\) \u3068, \\(xy\\) \u5e73\u9762\u4e0a\u306e\u5186 \\(C : \\ x^2 +(y-2)^2 = 1\\) \u306b\u5c5e\u3059\u308b\u70b9 Q \\(( \\cos \\theta , 2 +\\sin \\ … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n