\u5b9f\u6570 \\(\\theta\\) \u304c\u52d5\u304f\u3068\u304d, \\(xy\\) \u5e73\u9762\u4e0a\u306e\u52d5\u70b9 P \\(( 0 , \\sin \\theta )\\) \u304a\u3088\u3073 Q \\(( 8 \\cos \\theta , 0 )\\) \u3092\u8003\u3048\u308b.\r\n\\(\\theta\\) \u304c \\(0 \\leqq \\theta \\leqq \\dfrac{\\pi}{2}\\) \u306e\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d, \u5e73\u9762\u5185\u3067\u7dda\u5206 PQ \u304c\u901a\u904e\u3059\u308b\u90e8\u5206\u3092 \\(D\\) \u3068\u3059\u308b. \\(D\\) \u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u7acb\u4f53\u306e\u4f53\u7a4d \\(V\\) \u3092\u6c42\u3081\u3088.<\/p>\r\n
\u7dda\u5206 PQ \u306f,\r\n\\[\r\n\\dfrac{x}{8 \\cos \\theta} +\\dfrac{y}{\\sin \\theta} = 1 \\quad ( x \\geqq 0 , \\ y \\geqq 0 )\r\n\\]\r\n\u3068\u8868\u305b\u308b. \\(x=k \\ ( 0 \\leqq k \\leqq 8 )\\) \u3068\u56fa\u5b9a\u3059\u308b\u3068\r\n\\[\r\ny = \\sin \\theta -\\dfrac{k \\sin \\theta}{8 \\cos \\theta} = \\sin \\theta -\\dfrac{k}{8} \\tan \\theta\n\\]\r\n\\(y \\geqq 0\\) \u306a\u306e\u3067, \\(\\theta \\ \\left( 0 \\leqq \\theta \\leqq \\dfrac{\\pi}{2} \\right)\\) \u306e\u53d6\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u306f\r\n\\[\\begin{align}\r\n\\sin \\theta \\left( 1-\\dfrac{k}{8 \\cos \\theta} \\right) & \\geqq 0 \\\\\r\n\\text{\u2234} \\quad \\cos \\theta \\geqq \\dfrac{k}{8} \\quad & ( \\ \\text{\u2235} \\ \\sin \\theta \\geqq 0 \\ )\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n0 \\leqq \\theta \\leqq \\theta _ 0 \\quad \\left( \\cos \\theta _ 0 = \\dfrac{k}{8} \\right)\n\\]\r\n\\(\\theta\\) \u304c\u3053\u306e\u7bc4\u56f2\u3092\u52d5\u304f\u3068\u304d\u306e \\(y\\) \u306e\u53d6\u308a\u3046\u308b\u5024\u306e\u7bc4\u56f2\u3092\u8003\u3048\u308b.
\r\n\\(y = f( \\theta )\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nf'( \\theta ) & = \\cos \\theta +\\dfrac{k}{8 \\cos^2 \\theta} =\\dfrac{8 \\cos^3 \\theta -k}{8 \\cos^2 \\theta} \\\\\r\n& = \\dfrac{\\left( 2\\cos \\theta -k^{\\frac{1}{3}} \\right) \\left( 4\\cos^2 \\theta +2 k^{\\frac{1}{3}} \\cos \\theta +k^{\\frac{2}{3}} \\right)}{8 \\cos^2 \\theta}\n\\end{align}\\]\r\n\\(f'( \\theta ) =0\\) \u3092\u89e3\u304f\u3068\r\n\\[\r\n\\cos \\theta = \\dfrac{k^{\\frac{1}{3}}}{2}\n\\]\r\n\u3053\u308c\u3092\u6e80\u305f\u3059 \\(\\theta\\) \u3092 \\(\\alpha\\) \u3068\u304a\u3051\u3070, \\(f( \\theta )\\) \u306e\u5897\u6e1b\u8868\u306f\u4e0b\u56f3\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} \\theta & 0 & \\cdots & \\alpha & \\cdots & \\theta _ 0 \\\\ \\hline\r\nf'( \\theta ) & & + & 0 & - & \\\\ \\hline f( \\theta ) & 0 & \\nearrow & \\text{\u6700\u5927} & \\searrow & 0 \\end{array}\r\n\\]\r\n\\[\\begin{align}\r\n\\sin \\alpha & = \\sqrt{1-\\cos^2 \\alpha} =\\sqrt{1-\\dfrac{k^{\\frac{2}{3}}}{4}} , \\\\\r\n\\tan \\alpha & = \\dfrac{\\sin \\alpha}{\\cos \\alpha} =\\dfrac{2}{k^{\\frac{1}{3}}} \\sqrt{1-\\dfrac{k^{\\frac{2}{3}}}{4}}\n\\end{align}\\]\r\n\u306a\u306e\u3067,\r\n\\[\\begin{align}\r\nf( \\alpha ) & = \\sqrt{1-\\dfrac{k^{\\frac{2}{3}}}{4}} -\\dfrac{k}{8} \\cdot \\dfrac{2}{k^{\\frac{1}{3}}} \\sqrt{1-\\dfrac{k^{\\frac{2}{3}}}{4}} \\\\\r\n& = \\left( 1-\\dfrac{k^{\\frac{2}{3}}}{4} \\right)^{\\frac{3}{2}}\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n0 \\leqq f( \\theta ) \\leqq \\left( 1-\\dfrac{k^{\\frac{2}{3}}}{4} \\right)^{\\frac{3}{2}}\n\\]\r\n\u4ee5\u4e0a\u3088\u308a, \u6c42\u3081\u308b\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\\begin{align}\r\nV & = \\pi \\displaystyle\\int _ 0^8 \\left\\{ f( \\theta ) \\right\\}^2 \\, dk =\\pi \\displaystyle\\int _ 0^8 \\left( 1-\\dfrac{k^{\\frac{2}{3}}}{4} \\right)^3 \\, dk \\\\\r\n& = \\pi \\displaystyle\\int _ 0^8 \\left( 1 -\\dfrac{3}{4}k^{\\frac{2}{3}} +\\dfrac{3}{16}k^{\\frac{4}{3}} -\\dfrac{1}{64}k^2 \\right) \\, dk \\\\\r\n& = \\pi \\left[ k -\\dfrac{3}{4} \\cdot \\dfrac{3}{5}k^{\\frac{5}{3}} +\\dfrac{3}{16} \\cdot \\dfrac{3}{7}k^{\\frac{7}{3}} -\\dfrac{1}{64} \\cdot \\dfrac{1}{3}k^3 \\right] _ 0^8 \\\\\r\n& = \\pi \\left( 8 -\\dfrac{9 \\cdot 2^5}{20} +\\dfrac{9 \\cdot 2^7}{16 \\cdot 7} -\\dfrac{2^9}{64 \\cdot 3} \\right) \\\\\r\n& = \\pi \\left( 8 -\\dfrac{72}{5} +\\dfrac{72}{7} -\\dfrac{8}{3} \\right) \\\\\r\n& = 8 \\pi \\left( 1 -\\dfrac{9}{5} +\\dfrac{9}{7} -\\dfrac{1}{3} \\right) \\\\\r\n& = \\underline{\\dfrac{128 \\pi}{105}}\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\u5b9f\u6570 \\(\\theta\\) \u304c\u52d5\u304f\u3068\u304d, \\(xy\\) \u5e73\u9762\u4e0a\u306e\u52d5\u70b9 P \\(( 0 , \\sin \\theta )\\) \u304a\u3088\u3073 Q \\(( 8 \\cos \\theta , 0 )\\) \u3092\u8003\u3048\u308b. \\(\\theta\\) … \u7d9a\u304d\u3092\u8aad\u3080