(1)<\/strong>\u3000\\(a\\) \u304a\u3088\u3073\u70b9 Q \u306e \\(x\\) \u5ea7\u6a19\u3092 \\(p\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n (2)<\/strong>\u3000\u3053\u306e \\(2\\) \u3064\u306e\u66f2\u7dda\u3068 \\(x\\) \u8ef8\u3067\u56f2\u307e\u308c\u308b\u56f3\u5f62\u3092, \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u7acb\u4f53\u306e\u4f53\u7a4d\u3092 \\(p\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n (3)<\/strong>\u3000(2)<\/strong> \u3067\u5f97\u3089\u308c\u308b\u7acb\u4f53\u306e\u4f53\u7a4d\u304c \\(2 \\pi\\) \u306b\u306a\u308b\u3068\u304d\u306e \\(p\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(C : \\ y = a x^p\\) , \\(D : \\ y = \\log x\\) \u3068\u304a\u304f. (2)<\/strong><\/p>\r\n \\(C\\) \u3068 \\(D\\) \u306f\u4e0a\u56f3\u306e\u3088\u3046\u306a\u4f4d\u7f6e\u95a2\u4fc2\u306b\u3042\u308b\u306e\u3067, \u6c42\u3081\u308b\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\r\nV = \\pi \\displaystyle\\int _ 0^t \\left( a x^p \\right)^2 \\, dx -\\pi \\displaystyle\\int _ 1^t ( \\log x )^2 \\, dx\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\n\\displaystyle\\int ( \\log x )^2 \\, dx & = x ( \\log x )^2 -2 \\displaystyle\\int x \\cdot \\dfrac{\\log x}{x} \\, dx \\\\\r\n& = x ( \\log x )^2 -2x \\log x +2 \\displaystyle\\int x \\cdot \\dfrac{1}{x} \\, dx \\\\\r\n& = x ( \\log x )^2 -2x \\log x +2x +C , \\\\\r\n\\displaystyle\\int \\left( a x^p \\right)^2 \\, dx & = \\dfrac{a^2 x^{2p+1}}{2p+1} +C\r\n\\end{align}\\]\r\n\u305f\u3060\u3057, \\(C\\) \u306f\u7a4d\u5206\u5b9a\u6570. (3)<\/strong><\/p>\r\n \\(V = 2 \\pi\\) \u306a\u306e\u3067, (2)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\\begin{align}\r\n1-2p & = 0 \\\\\r\n\\text{\u2234} \\quad p & = \\underline{\\dfrac{1}{2}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a\\) \u3092\u6b63\u306e\u5b9f\u6570\u3068\u3057, \\(p\\) \u3092\u6b63\u306e\u6709\u7406\u6570\u3068\u3059\u308b. \u5ea7\u6a19\u5e73\u9762\u4e0a\u306e \\(2\\) \u3064\u306e\u66f2\u7dda \\(y = ax^p \\ ( x \\gt 0 )\\) \u3068 \\(y = \\log x \\ ( x \\gt 0 )\\) \u3092\u8003 … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
\r\n\u305d\u308c\u305e\u308c\u5fae\u5206\u3059\u308b\u3068\r\n\\[\r\ny' = ap x^{p-1} , \\quad y' = \\dfrac{1}{x}\r\n\\]\r\nQ \u306e \\(x\\) \u5ea7\u6a19\u3092 \\(t\\) \u3068\u304a\u304f\u3068\r\n\\[\r\n\\left\\{ \\begin{array}{ll} a t^p = \\log t & ... [1] \\\\ ap t^{p-1} = \\dfrac{1}{t} & ... [2] \\end{array} \\right.\r\n\\]\r\n[2] \u3088\u308a\r\n\\[\r\na = \\dfrac{1}{p t^p}\r\n\\]\r\n\u3053\u308c\u3092 [1] \u306b\u4ee3\u5165\u3057\u3066\r\n\\[\\begin{align}\r\n\\dfrac{1}{p t^p} \\cdot t^p & = \\log t \\\\\r\n\\text{\u2234} \\quad t & = e^{\\frac{1}{p}}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, Q \u306e \\(x\\) \u5ea7\u6a19\u306f, \\(\\underline{e^{\\frac{1}{p}}}\\) .
\r\n\u307e\u305f, \\(t^p = e\\) \u306a\u306e\u3067\r\n\\[\r\na = \\underline{\\dfrac{1}{pe}}\r\n\\]\r\n![\"tkr20150301\"](\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/tkr20150301.svg\")
\r\n
\r\n\u3053\u308c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nV & = \\dfrac{\\pi a^2 t^{2p+1}}{2p+1} -\\pi \\left\\{ t ( \\log t )^2 -2t \\log t +2t \\right\\} +2 \\pi \\\\\r\n& = \\dfrac{\\frac{1}{p^2 e^2} \\cdot e^{2 +\\frac{1}{p}}}{2p+1} -\\pi e^{\\frac{1}{p}} \\left( \\dfrac{1}{p^2} -\\dfrac{2}{p} +2 \\right) +2\\pi \\\\\r\n& = \\dfrac{\\pi e^{\\frac{1}{p}}}{p^2 (2p+1)} \\left\\{ 1 -(2p+1)( 2p^2 -2p +1 ) \\right\\} +2 \\pi \\\\\r\n& = \\dfrac{\\pi e^{\\frac{1}{p}}}{p^2 (2p+1)} \\cdot 2 t^2 (1-2p) +2 \\pi \\\\\r\n& = \\underline{2 \\pi \\left\\{ 1 +\\dfrac{e^{\\frac{1}{p}} (1-2p)}{1+2p} \\right\\}}\r\n\\end{align}\\]\r\n