(1)<\/strong>\u3000\\(a , b , r\\) \u306e\u5024\u3092\u305d\u308c\u305e\u308c\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\u9023\u7acb\u4e0d\u7b49\u5f0f\r\n\\[\r\na \\leqq x \\leqq 1 , \\quad 0 \\leqq y \\leqq \\sqrt{2} (x-1)^2 , \\quad x^2 +y^2 \\geqq r^2\r\n\\]\r\n\u306e\u8868\u3059\u9818\u57df\u3092, \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u4e0e\u3048\u3089\u308c\u305f\u653e\u7269\u7dda\u3092 \\(C\\) , \u5186\u3092 \\(D\\) \u3068\u3059\u308b. (2)<\/strong><\/p>\r\n \\(x\\) \u8ef8, \u76f4\u7dda \\(x = a\\) , \\(C\\) \u306b\u56f2\u307e\u308c\u305f\u9818\u57df\u3092 \\(R_1\\) , \\(x\\) \u8ef8, \u76f4\u7dda \\(x = a\\) , \\(D\\) \u306b\u56f2\u307e\u308c\u305f\u9818\u57df\u3092 \\(R_2\\) \u3068\u3059\u308b.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
![\"\"](\"https:\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/osr20160301.svg\")
<\/p>\r\n
\r\n\\(D\\) \u306e\u4e2d\u5fc3\u3067\u3042\u308b\u539f\u70b9 O \u306f, \\(C\\) \u306e\u4e0b\u65b9\u306b\u3042\u308b\u306e\u3067, \u6761\u4ef6\u3092\u307f\u305f\u3059\u306e\u306f, \\(C\\) \u3068\r\n\\(D\\) \u304c\u63a5\u3059\u308b\u3068\u304d\u3067\u3042\u308b.
\r\n\\(y = \\sqrt{2} (x-1)^2\\) \u3088\u308a\r\n\\[\r\ny' = 2 \\sqrt{2} (x-1)\r\n\\]\r\n\u306a\u306e\u3067, \u70b9 A \\((a,b)\\) \u306b\u304a\u3051\u308b\u63a5\u7dda \\(\\ell _\\text{A}\\) \u306e\u50be\u304d\u306f\r\n\\[\r\n2 \\sqrt{2} (a-1) \\quad ... [1]\r\n\\]\r\n\u307e\u305f, OA \u306e\u50be\u304d\u306f\r\n\\[\r\n\\dfrac{b}{a} = \\dfrac{\\sqrt{2} (a-1)^2}{a} \\quad ... [2]\r\n\\]\r\n\\(\\ell _\\text{A} \\perp \\text{OA}\\) \u306a\u306e\u3067, [1] [2] \u3088\u308a\r\n\\[\\begin{align}\r\n2 \\sqrt{2} (a-1) \\cdot \\dfrac{\\sqrt{2} (a-1)^2}{a} & = -1 \\\\\r\n4 (a-1)^3 +a & = 0 \\\\\r\n4a^3 -12a^2 +13a -4 & = 0 \\\\\r\n(2a-1) \\underline{( 2a^2 -5a +4 )} _{[3]} & = 0 \\\\\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \\([3] = 2 \\left( a -\\dfrac{5}{4} \\right)^2 +\\dfrac{7}{8} \\gt 0\\) \u3060\u304b\u3089\r\n\\[\r\na = \\underline{\\dfrac{1}{2}}\r\n\\]\r\n\u307e\u305f\r\n\\[\r\nb = \\sqrt{2} \\left( \\dfrac{1}{2} -1 \\right)^2 = \\underline{\\dfrac{\\sqrt{2}}{4}}\r\n\\]\r\n\u3055\u3089\u306b\r\n\\[\\begin{align}\r\nr & = \\sqrt{a^2 +b^2} = \\sqrt{\\dfrac{1}{4} +\\dfrac{1}{8}} \\\\\r\n& = \\sqrt{\\dfrac{3}{8}} = \\underline{\\dfrac{\\sqrt{6}}{4}}\r\n\\end{align}\\]\r\n![\"\"](\"https:\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/osr20160302.svg\")
<\/p>\r\n
\r\n\\(R_1 , R_2\\) \u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b\u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u7acb\u4f53\u306e\u4f53\u7a4d\u3092, \u305d\u308c\u305e\u308c \\(V_1 , V_2\\) \u3068\u304a\u304f.\r\n\\[\\begin{align}\r\nV_1 & = \\pi \\displaystyle\\int _{\\frac{1}{2}}^{1} 2 (x-1)^4 \\, dx \\\\\r\n& = 2 \\pi \\left[ \\dfrac{(x-1)^5}{5} \\right] _{\\frac{1}{2}}^{1} \\\\\r\n& = \\dfrac{\\pi}{80}\r\n\\end{align}\\]\r\n\u307e\u305f\r\n\\[\\begin{align}\r\nV_2 & = \\pi \\displaystyle\\int _{\\frac{1}{2}}^{\\frac{\\sqrt{6}}{4}} \\left( \\dfrac{3}{8} -x^2 \\right) \\, dx \\\\\r\n& = \\pi \\left[ \\dfrac{3x}{8} -\\dfrac{x^3}{3} \\right] _{\\frac{1}{2}}^{\\frac{\\sqrt{6}}{4}} \\\\\r\n& = \\pi \\left( \\dfrac{3 \\sqrt{6}}{32} -\\dfrac{\\sqrt{6}}{32} \\right) -\\pi \\left( \\dfrac{3}{16} -\\dfrac{1}{24} \\right) \\\\\r\n& = \\dfrac{\\sqrt{6} \\pi}{16} -\\dfrac{7 \\pi}{48}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\\begin{align}\r\nV & = V_1 -V_2 \\\\\r\n& = \\left( \\dfrac{3 +35}{240} -\\dfrac{\\sqrt{6}}{16} \\right) \\pi \\\\\r\n& = \\underline{\\dfrac{38 -15 \\sqrt{6}}{240} \\pi}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u5ea7\u6a19\u5e73\u9762\u306b\u304a\u3044\u3066, \u539f\u70b9 O \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u534a\u5f84 \\(r\\) \u306e\u5186\u3068\u653e\u7269\u7dda \\(y = \\sqrt{2} (x-1)^2\\) \u306f, \u305f\u3060 \\(1\\) \u3064\u306e\u5171\u6709\u70b9 \\(( a , b )\\) \u3092\u3082\u3064\u3068\u3059\u308b. (1)\u3000\\(a … \u7d9a\u304d\u3092\u8aad\u3080