(1)<\/strong>\u3000D \u306e\u6982\u5f62\u3092\u63cf\u304d, \u305d\u306e\u9762\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\u76f4\u7dda \\(x+y=a\\) \u3092\u8ef8\u3068\u3057\u3066, D \u3092 \\(1\\) \u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u56f3\u5f62\u306e\u4f53\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(\\ell : \\ x+y=a\\) , \\(C : \\ \\sqrt{x} +\\sqrt{y} =\\sqrt{a}\\) \u3068\u304a\u304f. \u307e\u305f, D \u306e\u9762\u7a4d \\(S\\) \u306f\r\n\\[\\begin{align}\r\nS & = \\displaystyle\\int _ 0^a \\{ x -( x -2\\sqrt{a} \\sqrt{x} +a ) \\} \\, dx \\\\\r\n& = \\left[ 2\\sqrt{a} \\cdot \\dfrac{2 x^{\\frac{3}{2}}}{3} -ax \\right] _ 0^a \\\\\r\n& =\\underline{\\dfrac{a^2}{3}}\r\n\\end{align}\\]\r\n (2)<\/strong><\/p>\r\n \\(\\ell\\) \u4e0a\u306e\u70b9 P \\((t,a-t) \\ ( 0 \\lt t \\lt a )\\) \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u5186\u3092\u5e95\u9762, \u9ad8\u3055 \\(\\sqrt{2} dt\\) \u306e\u5fae\u5c0f\u306a\u5186\u67f1\u306e\u4f53\u7a4d \\(dV\\) \u3092\u8003\u3048\u308c\u3070\u3088\u3044.
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\\(C\\) \u306e\u5f0f\u3088\u308a, \\(0 \\lt x \\lt a , \\ 0 \\lt y \\lt a\\) \u3067\u3042\u308a\r\n\\[\\begin{align}\r\ny & = \\left( \\sqrt{a} -\\sqrt{x} \\right)^2 \\\\\r\n& = x -2\\sqrt{a} \\sqrt{x} +a\r\n\\end{align}\\]\r\n\u3053\u308c\u3092\u5fae\u5206\u3059\u308b\u3068\r\n\\[\\begin{align}\r\ny' & = 1 -\\dfrac{2 \\sqrt{a}}{2\\sqrt{x}} = -\\dfrac{\\sqrt{a} -\\sqrt{x}}{\\sqrt{x}} \\lt 0 , \\\\\r\ny'' & = \\dfrac{3}{2 x^{\\frac{3}{2}}} \\gt 0\r\n\\end{align}\\]\r\n\u306a\u306e\u3067, \\(C\\) \u306f \\(0 \\lt x \\lt a\\) \u306b\u304a\u3044\u3066, \u4e0b\u306b\u51f8\u3067\u5358\u8abf\u6e1b\u5c11\u3059\u308b\u66f2\u7dda\u3067\u3042\u308b.
\r\n\u3055\u3089\u306b\r\n\\[\r\n\\displaystyle\\lim _ {x \\rightarrow +0} y' = -\\infty , \\ \\displaystyle\\lim _ {x \\rightarrow a-0} y' = 0\r\n\\]\r\n\u3067\u3042\u308b\u3053\u3068\u306b\u6ce8\u610f\u3059\u308c\u3070, D \u306e\u6982\u5f62\u306f\u4e0b\u56f3\u659c\u7dda\u90e8\u306e\u3088\u3046\u306b\u306a\u308b.<\/p>\r\n![\"waseda_r_2008_01_01\"](\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/waseda_r_2008_01_01.png\")
\r\n
\r\nP \u3092\u901a\u308b \\(\\ell\\) \u306b\u5782\u76f4\u306a\u76f4\u7dda\u306e\u65b9\u7a0b\u5f0f\u306f\r\n\\[\r\ny = x +a- 2t\r\n\\]\r\n\u3053\u308c\u3068 \\(C\\) \u3068\u306e\u4ea4\u70b9\u3092\u8003\u3048\u308b\u3068\r\n\\[\\begin{align}\r\nx +a-2t & = x -2\\sqrt{a} \\sqrt{x} +a \\\\\r\n2 \\sqrt{a} \\sqrt{x} & = 2t \\\\\r\n\\text{\u2234} \\quad x & = \\dfrac{t^2}{a}\r\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\\begin{align}\r\ndV & = \\left\\{ \\sqrt{2} \\left( t -\\dfrac{t^2}{a} \\right) \\right\\}^2 \\pi \\cdot \\sqrt{2} dt \\\\\r\n& = \\dfrac{2 \\sqrt{2} \\pi}{a^2} ( at-t^2 )^2 dt\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\\begin{align}\r\nV & = \\displaystyle\\int _ 0^a \\dfrac{2 \\sqrt{2} \\pi}{a^2} ( at-t^2 )^2 \\, dt \\\\\r\n& = \\dfrac{2 \\sqrt{2} \\pi}{a^2} \\left[ \\dfrac{a^2 t^3}{3} -\\dfrac{a t^4}{2} +\\dfrac{t^5}{5} \\right] _ 0^a \\\\\r\n& = \\dfrac{2 \\sqrt{2}}{a^2} \\cdot \\dfrac{a^5}{30} \\\\\r\n& = \\underline{\\dfrac{\\sqrt{2} a^3 \\pi}{15}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a\\) \u3092\u6b63\u306e\u5b9a\u6570\u3068\u3059\u308b. \\(xy\\) \u5ea7\u6a19\u5e73\u9762\u306b\u304a\u3044\u3066, \u66f2\u7dda \\(\\sqrt{x} +\\sqrt{y} = \\sqrt{a}\\) \u3068, \u76f4\u7dda \\(x+y=a\\) \u3068\u3067\u56f2\u307e\u308c\u305f\u90e8\u5206\u3092 D \u3068\u304a\u304f. \u4ee5\u4e0b\u306e\u554f\u306b\u7b54\u3048 … \u7d9a\u304d\u3092\u8aad\u3080