\\(u\\) \u3092\u5b9f\u6570\u3068\u3059\u308b. \u5ea7\u6a19\u5e73\u9762\u4e0a\u306e \\(2\\) \u3064\u306e\u653e\u7269\u7dda\r\n\\[\\begin{align}\r\nC _ 1 : \\quad & y = -x^2+1 \\\\\r\nC _ 2 : \\quad & y = (x-u)^2+u\r\n\\end{align}\\]\r\n\u3092\u8003\u3048\u308b. \\(C _ 1\\) \u3068 \\(C _ 2\\) \u304c\u5171\u6709\u70b9\u3092\u3082\u3064\u3088\u3046\u306a \\(u\\) \u306e\u5024\u306e\u7bc4\u56f2\u306f, \u3042\u308b\u5b9f\u6570 \\(a , b\\) \u306b\u3088\u308a, \\(a \\leqq x \\leqq b\\) \u3068\u8868\u3055\u308c\u308b.<\/p>\r\n
(1)<\/strong>\u3000\\(a , b\\) \u306e\u5024\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\\(u\\) \u304c \\(a \\leqq x \\leqq b\\) \u3092\u307f\u305f\u3059\u3068\u304d, \\(C _ 1\\) \u3068 \\(C _ 2\\) \u306e\u5171\u6709\u70b9\u3092 \\(\\text{P} {} _ 1 \\ ( x _ 1 , y _ 1 )\\) , \\(\\text{P} {} _ 2 \\ ( x _ 2 , y _ 2 )\\) \u3068\u3059\u308b. \u305f\u3060\u3057, \u5171\u6709\u70b9\u304c \\(1\\) \u70b9\u306e\u307f\u306e\u3068\u304d\u306f, \\(\\text{P} {} _ 1\\) \u3068 \\(\\text{P} {} _ 2\\) \u306f\u4e00\u81f4\u3057, \u3068\u3082\u306b\u305d\u306e\u5171\u6709\u70b9\u3092\u8868\u3059\u3068\u3059\u308b.\r\n\\[\r\n2 \\left| x _ 1 y _ 2 -x _ 2 y _ 1 \\right|\r\n\\]\r\n\u3092 \\(u\\) \u306e\u5f0f\u3067\u8868\u305b.<\/p><\/li>\r\n (3)<\/strong>\u3000(2)<\/strong> \u3067\u5f97\u3089\u308c\u308b \\(u\\) \u306e\u5f0f\u3092 \\(f(u)\\) \u3068\u3059\u308b. \u5b9a\u7a4d\u5206\r\n\\[\r\nI = \\displaystyle\\int _ a^b f(u) \\, du\r\n\\]\r\n\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \\(C _ 1\\) \u3068 \\(C _ 2\\) \u304b\u3089 \\(y\\) \u3092\u6d88\u53bb\u3059\u308b\u3068\r\n\\[\\begin{align}\r\n-x^2 + 1 = (x-u)^2 & +u \\\\\r\n2x^2 -2ux +u^2 +u -1 & = 0 \\quad ... [1]\r\n\\end{align}\\]\r\n\u3053\u306e \\(x\\) \u306e \\(2\\) \u6b21\u65b9\u7a0b\u5f0f\u304c, \u5b9f\u6570\u89e3\u3092\u3082\u3066\u3070\u3088\u3044\u306e\u3067, \u5224\u5225\u5f0f \\(D\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\n\\dfrac{D}{4} & = u^2 -2 \\left( u^2 +u -1 \\right) \\geqq 0 \\\\\r\n& u^2 +2u -2 \\leqq 0 \\\\\r\n\\text{\u2234} \\quad & -1 -\\sqrt{3} \\leqq u \\leqq -1 +\\sqrt{3}\r\n\\end{align}\\]\r\n\u3088\u3063\u3066\r\n\\[\r\na = \\underline{-1 -\\sqrt{3}} , \\quad b = \\underline{-1 +\\sqrt{3}}\r\n\\]\r\n (2)<\/strong><\/p>\r\n \u89e3\u3068\u4fc2\u6570\u306e\u95a2\u4fc2\u304b\u3089, [1] \u3088\u308a\r\n\\[\r\nx _ 1 +x _ 2 = u , \\ x _ 1 x _ 2 = \\dfrac{1}{2} \\left( u^2 +u -1 \\right) \\quad ... [2]\r\n\\]\r\n\\(\\text{P} {} _ 1 , \\text{P} {} _ 2\\) \u306f\u3068\u3082\u306b \\(C _ 1\\) \u4e0a\u306e\u70b9\u306a\u306e\u3067\r\n\\[\r\ny _ 1 = -{x _ 1}^2 +1 , \\ y _ 2 = -{x _ 2}^2 +1\r\n\\]\r\n\u3053\u308c\u3068 [2] \u3092\u7528\u3044\u308c\u3070, \u6c42\u3081\u308b\u5024\u306f\r\n\\[\\begin{align}\r\n2 \\left| x _ 1 y _ 2 -x _ 2 y _ 1 \\right| & = 2 \\left| x _ 1 \\left( -{x _ 1}^2 +1 \\right) -x _ 2 \\left( -{x _ 2}^2 +1 \\right) \\right| \\\\\r\n& = 2 \\left| \\left( x _ 1 x _ 2 +1 \\right) \\left( x _ 1 -x _ 2 \\right) \\right| \\\\\r\n& = \\left| u^2 +u +1 \\right| \\sqrt{u^2 -2 \\left( u^2 +u -1 \\right)} \\\\\r\n& = \\underline{\\left( u^2 +u +1 \\right) \\sqrt{-u^2 -2u +2}}\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \\(u^2 +u +1 = \\left( u +\\dfrac{1}{2} \\right)^2 +\\dfrac{3}{4} \\gt 0\\) \u3092\u7528\u3044\u305f.<\/p>\r\n (3)<\/strong><\/p>\r\n \\[\r\nf(u) = \\left( u^2 +u +1 \\right) \\sqrt{3 -(u+1)^2}\r\n\\]\r\n\u3053\u3053\u3067, \\(t = u+1\\) \u3068\u304a\u304f\u3068, \\(u = t-1\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\nf(u) & = \\left\\{ (t-1)^2 +(t-1) +1 \\right\\} \\sqrt{3-t^2} \\\\\r\n& = \\left( t^2 -t +1 \\right) \\sqrt{3-t^2}\r\n\\end{align}\\]\r\n\u307e\u305f, \\(du = dt\\) \u3067\u3042\u308a,\r\n\\[\r\nu : \\ -1 -\\sqrt{3} \\rightarrow -1 +\\sqrt{3} \\text{\u306e\u3068\u304d, } \\quad t : \\ -\\sqrt{3} \\rightarrow +\\sqrt{3}\r\n\\]\r\n\u306a\u306e\u3067, \u7f6e\u63db\u7a4d\u5206\u3059\u308b\u3068\r\n\\[\r\nI = \\displaystyle\\int _ {-\\sqrt{3}}^{\\sqrt{3}} \\left( t^2 -t +1 \\right) \\sqrt{3-t^2} \\, dt\r\n\\]\r\n\u5947\u95a2\u6570, \u5076\u95a2\u6570\u306e\u6027\u8cea\u3092\u7528\u3044\u308c\u3070\r\n\\[\r\nI = 2 \\displaystyle\\int _ {0}^{\\sqrt{3}} \\left( t^2 +3 \\right) \\sqrt{3-t^2} \\, dt\r\n\\]\r\n\u3055\u3089\u306b, \\(t = \\sqrt{3} \\sin \\theta \\quad \\left( 0 \\leqq \\theta \\leqq \\dfrac{\\pi}{2} \\right)\\) \u3068\u304a\u304f\u3068\r\n\\[\r\ndt = \\sqrt{3} \\cos \\theta \\, d \\theta\r\n\\]\r\n\u3067\u3042\u308a\r\n\\[\r\nt : \\ -\\sqrt{3} \\rightarrow +\\sqrt{3} \\text{\u306e\u3068\u304d, } \\quad \\theta : \\ 0 \\rightarrow \\dfrac{\\pi}{2}\r\n\\]\r\n\u306a\u306e\u3067, \u7f6e\u63db\u7a4d\u5206\u3059\u308b\u3068\r\n\\[\\begin{align}\r\nI & = 6 \\displaystyle\\int _ {0}^{\\frac{\\pi}{2}} \\left( 3 \\sin^2 \\theta +1 \\right) \\cos^2 \\theta \\, d \\theta \\\\\r\n& = 6 \\displaystyle\\int _ {0}^{\\frac{\\pi}{2}} \\left( 3 \\sin^2 \\theta \\cos^2 \\theta +\\cos^2 \\theta \\right) \\, d \\theta \\\\\r\n& = 6 \\displaystyle\\int _ {0}^{\\frac{\\pi}{2}} \\left( \\dfrac{3}{4} \\sin^2 2 \\theta +\\dfrac{1 +\\cos 2 \\theta}{2} \\right) \\, d \\theta \\\\\r\n& = 3 \\displaystyle\\int _ {0}^{\\frac{\\pi}{2}} \\left\\{ \\dfrac{3}{4} \\left( 1 -\\cos 4 \\theta \\right) +1 +\\cos 2 \\theta \\right\\} \\, d \\theta \\\\\r\n& = 3 \\left[ -\\dfrac{3}{16} \\sin 4 \\theta +\\dfrac{1}{2} \\sin 2 \\theta + \\dfrac{7 \\theta }{4} \\right] _ {0}^{\\frac{\\pi}{2}} \\\\\r\n& = 3 \\cdot \\dfrac{7 \\pi}{8} \\\\\r\n& = \\underline{\\dfrac{21 \\pi}{8}}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(u\\) \u3092\u5b9f\u6570\u3068\u3059\u308b. \u5ea7\u6a19\u5e73\u9762\u4e0a\u306e \\(2\\) \u3064\u306e\u653e\u7269\u7dda \\[\\begin{align} C _ 1 : \\quad & y = -x^2+1 \\\\ C _ 2 : \\quad & y = (x-u)^2+u \\ … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n