(1)<\/strong>\u3000\\(A\\) \u306e\u4f53\u7a4d\u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n (2)<\/strong>\u3000\u70b9 \\(( t , 0 ) \\ ( -a \\leqq t \\leqq a )\\) \u3092\u901a\u308a \\(x\\) \u8ef8\u3068\u5782\u76f4\u306a\u5e73\u9762\u306b\u3088\u308b \\(A\\) \u306e\u5207\u308a\u53e3\u306e\u9762\u7a4d\u3092 \\(S(t)\\) \u3068\u3059\u308b\u3068\u304d, \u4e0d\u7b49\u5f0f\r\n\\[\r\nS(t) \\leqq \\displaystyle\\int _ {-a}^a e^{-( s^2+t^2 )} \\, ds\r\n\\]\r\n\u3092\u793a\u305b.<\/p><\/li>\r\n (3)<\/strong>\u3000\u4e0d\u7b49\u5f0f\r\n\\[\r\n\\sqrt{\\pi \\left( 1 -e^{-a^2} \\right)} \\leqq \\displaystyle\\int _ {-a}^a e^{-x^2} \\, dx\r\n\\]\r\n\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n (1)<\/strong><\/p>\r\n \u6c42\u3081\u308b\u4f53\u7a4d \\(V\\) \u306f\r\n\\[\\begin{align}\r\nV & = 2 \\pi \\displaystyle\\int _ 0^a x e^{-x^2} \\, dx \\\\\r\n& = -\\pi \\displaystyle\\int _ 0^a e^{-x^2} (-x^2)' \\, dx \\\\\r\n& = -\\left[ e^{-x^2} \\right] _ 0^a \\\\\r\n& = \\underline{\\pi \\left( 1 -e^{-a^2} \\right)}\r\n\\end{align}\\]\r\n (2)<\/strong><\/p>\r\n \u5e73\u9762 \\(y = 0\\) \u304c \\(xz\\) \u5e73\u9762\u306b\u306a\u308b\u3088\u3046\u306b, \\(z\\) \u8ef8\u3092\u5b9a\u3081\u308b. (3)<\/strong><\/p>\r\n \\(V = \\displaystyle\\int _ {-a}^a S(t) \\, dt\\) \u306a\u306e\u3067, (2)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\nV & \\leqq \\displaystyle\\int _ {-a}^a \\left( e^{-t^2} \\displaystyle\\int _ {-a}^a e^{-s^2} \\, ds \\right) \\, dt \\\\\r\n& = \\left( \\displaystyle\\int _ {-a}^a e^{-s^2} \\, ds \\right) \\left( \\displaystyle\\int _ {-a}^a e^{-t^2} \\, dt \\right) \\\\\r\n& = \\left( \\displaystyle\\int _ {-a}^a e^{-x^2} \\, dx \\right)^2\r\n\\end{align}\\]\r\n\\(e^{-x^2} \\gt 0\\) \u3088\u308a, \\(\\displaystyle\\int _ {-a}^a e^{-x^2} \\, dx \\gt 0\\) \u306a\u306e\u3067, (1)<\/strong> \u306e\u7d50\u679c\u3092\u4ee3\u5165\u3057\u3066\r\n\\[\r\n\\sqrt{\\pi \\left( 1 -e^{-a^2} \\right)} \\leqq \\displaystyle\\int _ {-a}^a e^{-x^2} \\, dx\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\\(a \\gt 0\\) \u3068\u3059\u308b. \u66f2\u7dda \\(y = e^{-x^2}\\) \u3068 \\(x\\) \u8ef8, \\(y\\) \u8ef8, \u304a\u3088\u3073\u76f4\u7dda \\(x = a\\) \u3067\u56f2\u307e\u308c\u305f\u56f3\u5f62\u3092, \\(y\\) \u8ef8\u306e\u307e\u308f\u308a\u306b \\(1\\) \u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u56de … \u7d9a\u304d\u3092\u8aad\u3080
\r\n\r\n\u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n
![\"tok20150301\"](\"\/\/www.roundown.net\/nyushi\/wp-content\/uploads\/tok20150301.svg\")
\r\n
\r\n\\(A\\) \u306f \\(y\\) \u8ef8\u3092\u4e2d\u5fc3\u3068\u3057\u305f\u56de\u8ee2\u4f53\u306a\u306e\u3067, \\(xz\\) \u5e73\u9762\u3067, \u70b9 \\(( t , s )\\) \u306b\u304a\u3051\u308b \\(A\\) \u306e\u65ad\u9762\u306e \\(y\\) \u8ef8\u65b9\u5411\u306e\u9ad8\u3055\u306f, \u70b9 \\(( \\sqrt{s^2 +t^2} , 0 )\\) \u306e\u305d\u308c\u3068\u7b49\u3057\u304f\r\n\\[\r\ne^{-( s^2 +t^2 )}\r\n\\]\r\n\u3088\u3063\u3066\r\n\\[\r\nS(t) = \\displaystyle\\int _ {-\\sqrt{a^2 -t^2}}^{\\sqrt{a^2 -t^2}} e^{-( s^2 +t^2 )} \\, ds \\leqq \\displaystyle\\int _ {-a}^a e^{-( s^2 +t^2 )} \\, ds\r\n\\]\r\n