{"id":1993,"date":"2019-05-17T12:54:53","date_gmt":"2019-05-17T03:54:53","guid":{"rendered":"https:\/\/sasamath.com\/wp\/?p=1993"},"modified":"2019-09-05T19:59:43","modified_gmt":"2019-09-05T10:59:43","slug":"calculus-moments-and-centers-of-mass","status":"publish","type":"post","link":"https:\/\/sasamath.com\/blog\/articles\/calculus-moments-and-centers-of-mass\/","title":{"rendered":"\ubaa8\uba58\ud2b8\uc640 \uc9c8\ub7c9\uc911\uc2ec"},"content":{"rendered":"

\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ubaa8\uba58\ud2b8\uc640 \uc9c8\ub7c9\uc911\uc2ec\uc744 \uc218\ud559\uc801\uc73c\ub85c \uc815\uc758\ud558\uace0, \uc9c8\ub7c9\uc911\uc2ec\uc744 \uacc4\uc0b0\ud558\ub294 \uacf5\uc2dd\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \ub354\ubd88\uc5b4 \uc9c8\ub7c9\uc911\uc2ec\uc744 \uc774\uc6a9\ud558\uc5ec \ud68c\uc804\uccb4\uc758 \ubd80\ud53c\uc640 \ub113\uc774\ub97c \uc27d\uac8c \uacc4\uc0b0\ud560 \uc218 \uc788\ub294 \ud30c\ud478\uc2a4\uc758 \uc815\ub9ac\ub97c \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

<\/p>\n

\ubaa8\uba58\ud2b8\uc640 \uc9c8\ub7c9\uc911\uc2ec\uc758 \uc815\uc758<\/h3>\n

\ubaa8\uba58\ud2b8\uc640 \uc9c8\ub7c9\uc911\uc2ec\uc740 \uc138 \ub2e8\uacc4\ub85c \uc815\uc758\ud55c\ub2e4. \uba3c\uc800 \uc9c1\uc120 \uc704\uc5d0 \ub193\uc778 \uc720\ud55c \uac1c\uc758 \ubb3c\uccb4\uc5d0 \ub300\ud558\uc5ec \uc815\uc758\ud558\uace0, \ub2e4\uc74c\uc73c\ub85c \ud3c9\uba74\uc5d0 \ub193\uc778 \uc720\ud55c \uac1c\uc758 \ubb3c\uccb4\uc5d0 \ub300\ud558\uc5ec \uc815\uc758\ud55c \ub4a4, \ub9c8\uc9c0\ub9c9\uc73c\ub85c \ud3c9\uba74\uc5d0 \ub193\uc778 \ubb3c\uccb4(\uac01 \uc88c\ud45c\uc5d0\uc11c \ubc00\ub3c4\uac00 \ud568\uc218\ub85c \uc8fc\uc5b4\uc9c4)\uc5d0 \ub300\ud558\uc5ec \uc815\uc758\ud55c\ub2e4.<\/p>\n

(1) \uc9c1\uc120 \uc704\uc5d0 \ub193\uc778 \uc720\ud55c \uac1c\uc758 \ubb3c\uccb4<\/h4>\n

\uc544\ub798 \uadf8\ub9bc\ucc98\ub7fc \uc218\uc9c1\uc120 \uc704\uc5d0 \uc138 \ubb3c\uccb4\uac00 \ub193\uc5ec \uc788\ub2e4. \uac01 \ubb3c\uccb4\uc758 \uc704\uce58\ub294 \\(x_k\\)\uc774\uace0 \uc9c8\ub7c9\uc740 \\(m_k\\)\uc774\ub2e4. \uadf8\ub9ac\uace0 \uc6d0\uc810\uc5d0\ub294 \ubc1b\uce68\ub300\uac00 \ub193\uc5ec \uc788\ub2e4. (\ubb3c\uccb4\uc758 \ud06c\uae30\uac00 \ubb34\ucc99 \uc791\uc544\uc11c \ubb3c\uccb4\uc758 \uc704\uce58\uac00 \ud55c \uc810\uc73c\ub85c \ud45c\ud604\ub420 \uc218 \uc788\ub294 \uc774\uc0c1\uc801\uc778 \uc0c1\ud669\uc744 \uac00\uc815\ud558\uc790.)<\/p>\n

\"\"<\/p>\n

\uc6d0\uc810\uc744 \uae30\uc900\uc73c\ub85c \ud588\uc744 \ub54c \uac01 \ubb3c\uccb4\uac00 \uc9c0\ub81b\ub300\uc5d0 \ubbf8\uce58\ub294 \ud1a0\ud06c, \uc989 \uc9c0\ub81b\ub300\ub97c \ud68c\uc804\ud558\uac8c \ub9cc\ub4dc\ub294 \ud798\uc740 \\(g x_k m_k \\)\uc774\ub2e4. \uc5ec\uae30\uc11c \\(g\\)\ub294 \uc0c1\uc218\uc774\ub2e4. \uc774\ub54c \uacc4(system; \ubb3c\uccb4 \uc804\uccb4)\uac00 \uc9c0\ub81b\ub300\ub97c \ud68c\uc804\ud558\uac8c \ub9cc\ub4dc\ub294 \ud798\uc740
\n\\[g x_1 m_1 + gx_2 m_2 + gx_3 m_3 = g(x_1 m_1 + x_2 m_2 + x_3 m_3 )\\]
\n\uc774\ub2e4. \uc5ec\uae30\uc11c \\((x_1 m_1 + x_2 m_2 + x_3 m_3 )\\)\uc744 \uc6d0\uc810\uc5d0 \ub300\ud55c \uacc4\uc758 \ubaa8\uba58\ud2b8<\/span>(moment of the system about the origin)\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774 \uac12\uc774 \uc591\uc218\uc774\uba74 \uc9c0\ub81b\ub300\uc758 \uc624\ub978\ucabd \ubd80\ubd84\uc774 \ub0b4\ub824\uac00\uace0, \uc774 \uac12\uc774 \uc74c\uc218\uc774\uba74 \uc9c0\ub81b\ub300\uc758 \uc67c\ucabd \ubd80\ubd84\uc774 \ub0b4\ub824\uac04\ub2e4.<\/p>\n

\uc9c0\ub81b\ub300\uac00 \uc5b4\ub290 \ubd80\ubd84\ub3c4 \ub0b4\ub824\uac00\uc9c0 \uc54a\uace0 \uc218\ud3c9\ud558\uac8c \uc720\uc9c0\ub418\ub824\uba74 \ubc1b\uce68\ub300\ub97c \uc5b4\ub290 \uc704\uce58\uc5d0 \ub193\uc544\uc57c \ud560\uae4c? \uc544\ub798 \uadf8\ub9bc\uc5d0\uc11c\ucc98\ub7fc \uadf8 \uc704\uce58\ub97c \\(\\overline{x}\\)\ub77c\uace0 \ud558\uc790.<\/p>\n

\"\"<\/p>\n

\uadf8\ub7ec\uba74 \\(\\overline{x}\\)\uc5d0 \ub300\ud55c \uacc4\uc758 \ubaa8\uba58\ud2b8\ub294
\n\\[\\sum_{k=1}^{3} m_k (x_k – \\overline{x} )\\]
\n\uc774 \ub41c\ub2e4. \uc774 \uac12\uc774 \\(0\\)\uc774\uba74 \uc9c0\ub81b\ub300\ub294 \ud3c9\ud615\uc0c1\ud0dc\uac00 \ub418\ubbc0\ub85c
\n\\[\\sum_{k=1}^{3} m_k (x_k – \\overline{x} ) =0\\]
\n\uc989
\n\\[\\overline{x} = \\frac{\\sum x_k m_k }{\\sum m_k} = \\frac{\\text{(system moment about origin)}}{\\text{(system mass)}}\\]
\n\ub97c \uc5bb\ub294\ub2e4. \uc774\ub54c \\(\\overline{x}\\)\ub97c \uc9c8\ub7c9\uc911\uc2ec<\/span>(center of mass)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

(2) \ud3c9\uba74\uc5d0 \ub193\uc778 \uc720\ud55c \uac1c\uc758 \ubb3c\uccb4<\/h4>\n

\ud3c9\uba74\uc5d0\uc11c\ub3c4 \uc774\uc640 \ube44\uc2b7\ud55c \ubc29\ubc95\uc73c\ub85c \ubaa8\uba58\ud2b8\uc640 \uc9c8\ub7c9\uc911\uc2ec\uc744 \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4. \uc88c\ud45c\ud3c9\uba74\uc5d0 \\(n\\)\uac1c\uc758 \ubb3c\uccb4\uac00 \ub193\uc5ec \uc788\uace0, \uac01 \ubb3c\uccb4\uc758 \uc9c8\ub7c9\uacfc \uc704\uce58\ub294 \uac01\uac01 \\(m_k ,\\) \\((x_k ,\\, y_k ) \\)\ub77c\uace0 \ud558\uc790.<\/p>\n

\"\"<\/p>\n

\uc774 \uc0c1\ud669\uc5d0\uc11c\ub294 \\(x\\)\ucd95\uc5d0 \ub300\ud55c \ubaa8\uba58\ud2b8\uc640 \\(y\\)\ucd95\uc5d0 \ub300\ud55c \ubaa8\uba58\ud2b8\ub97c \uc0dd\uac01\ud560 \uc218 \uc788\ub2e4. \\(x\\)\ucd95\uc744 \uae30\uc900\uc73c\ub85c \ud588\uc744 \ub54c \ud55c \ubb3c\uccb4\uc758 \ud1a0\ud06c\ub294 \\(g y_k m_k\\)\uc774\ubbc0\ub85c \\(x\\)\ucd95\uc5d0 \ub300\ud55c \uacc4\uc758 \ud1a0\ud06c\ub294
\n\\[\\sum_{k=1}^{n} gy_k m_k \\]
\n\uc774\uba70, \\(x\\)\ucd95\uc5d0 \ub300\ud55c \uacc4\uc758 \ubaa8\uba58\ud2b8\ub294
\n\\[M_x = \\sum_{k=1}^{n} y_k m_k\\]
\n\uc774\ub2e4. \ub610\ud55c \uacc4\uc758 \uc9c8\ub7c9\uc740 \uac01 \ubb3c\uccb4\uc758 \uc9c8\ub7c9\uc758 \ud569\uacfc \uac19\uc73c\ubbc0\ub85c
\n\\[M = \\sum_{k=1}^{n} m_k\\]
\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uacc4\uc758 \uc9c8\ub7c9\uc911\uc2ec\uc758 \\(y\\)\uc88c\ud45c\ub294
\n\\[\\overline{y} = \\frac{M_x}{M} = \\frac{\\sum y_k m_k }{\\sum m_k}\\]
\n\uc774\ub2e4. \ube44\uc2b7\ud55c \ubc29\ubc95\uc73c\ub85c \\(y\\)\ucd95\uc5d0 \ub300\ud55c \uacc4\uc758 \ubaa8\uba58\ud2b8\ub294
\n\\[M_y = \\sum_{k=1}^{n} x_k m_k\\]
\n\uc774\uba70, \uacc4\uc758 \uc9c8\ub7c9\uc911\uc2ec\uc758 \\(x\\)\uc88c\ud45c\ub294
\n\\[\\overline{x} = \\frac{M_y}{M} = \\frac{\\sum x_k m_k }{\\sum m_k}\\]
\n\uc774\ub2e4.<\/p>\n

(3) \ud3c9\uba74\uc5d0 \ub193\uc778 \ubb3c\uccb4<\/h4>\n

\uc544\ub798 \uadf8\ub9bc\uc5d0\uc11c\ucc98\ub7fc \ud3c9\uba74\uc5d0 \uc587\uace0 \ud3c9\ud3c9\ud55c \ubb3c\uccb4\uac00 \ub193\uc5ec \uc788\ub2e4\uace0 \ud558\uc790. \ubb3c\uccb4\ub294 \ub9e4\ub044\ub7ec\uc6b4 \uace1\uc120\uc73c\ub85c \ub458\ub7ec\uc2f8\uc778 \ubaa8\uc591\uc774\ub2e4.<\/p>\n

\"\"<\/p>\n

\uc774 \ubb3c\uccb4\ub97c \\(y\\)\ucd95\uc5d0 \ud3c9\ud589\ud55c \uc9c1\uc120\uc73c\ub85c \uc798\ub77c\uc11c \uc5ec\ub7ec \uac1c\uc758 \ub760(strip)\uac00 \ub418\ub3c4\ub85d \ud558\uc790. [\ubb3c\ub860 \uc774 \ub760\ub294 \uc9c1\uc0ac\uac01\ud615 \ubaa8\uc591\uc774 \uc544\ub2c8\uc9c0\ub9cc, (\uad6c\ubd84\uad6c\uc801\ubc95\uc73c\ub85c \uc815\uc801\ubd84\uc744 \uacc4\uc0b0\ud560 \ub54c\ucc98\ub7fc) \ud3ed\uc774 \uc544\uc8fc \uc791\uc740 \uc9c1\uc0ac\uac01\ud615\uc774\ub77c\uace0 \uc0dd\uac01\ud574\ub3c4 \ubb34\ubc29\ud558\ub2e4.] \uac01 \ub760\uc758 \uc9c8\ub7c9\uc911\uc2ec\uc740 \\((\\tilde{x} ,\\, \\tilde{y} )\\)\uc774\uace0 \uc9c8\ub7c9\uc740 \\(\\Delta x\\)\ub77c\uace0 \ud558\uc790. \uac01 \ub760\ub97c \ud558\ub098\uc758 \ubb3c\uccb4\ub85c \uc0dd\uac01\ud558\uba74 \ubb3c\uccb4 \uc804\uccb4\uc758 \uc9c8\ub7c9\uc911\uc2ec\uc758 \uc88c\ud45c \\((\\overline{x} ,\\, \\overline{y} )\\)\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uad6c\ud560 \uc218 \uc788\ub2e4.
\n\\[\\overline{x} = \\frac{M_y}{M} = \\frac{\\sum \\tilde{x} \\Delta m}{\\sum \\Delta m} ,\\quad
\n\\overline{y} = \\frac{M_x}{M} = \\frac{\\sum \\tilde{y} \\Delta m}{\\sum \\Delta m} .\\]<\/p>\n

\uc5ec\uae30\uc11c \uac01 \ub760\uc758 \ud3ed\uc774 \ubb34\ud55c\ud788 \uc791\uc544\uc9c0\ub294 \uadf9\ud55c, \uc989 \\(\\Delta m \\,\\to\\,0\\)\uc778 \uadf9\ud55c\uc744 \uc0dd\uac01\ud558\uba74 \uc704 \uacf5\uc2dd\uc5d0\uc11c \uac01\uac01\uc758 \ud569\uc740 \uc801\ubd84\uc774 \ub418\ubbc0\ub85c, \ubb3c\uccb4 \uc804\uccb4\uc758 \uc9c8\ub7c9\uc911\uc2ec\uc758 \uc88c\ud45c\ub294 \ub2e4\uc74c\uacfc \uac19\uc774 \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\n

\\[\\overline{x} = \\frac{\\int \\tilde{x} \\,dm}{\\int dm} ,\\quad
\n\\overline{y} = \\frac{\\int \\tilde{x} \\,dm}{\\int dm} .\\]<\/p>\n

\uc9c0\uae08\uae4c\uc9c0 \uc0b4\ud3b4\ubcf8 \ub0b4\uc6a9\uc744 \uc815\ub9ac\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.\n<\/p>\n

\n

\uc815\uc758 1. (\ud3c9\uba74\uc5d0 \ub193\uc778 \ubb3c\uccb4\uc758 \ubaa8\uba58\ud2b8\uc640 \uc9c8\ub7c9\uc911\uc2ec)<\/span><\/p>\n

\\[\\begin{align}
\n\\text{Moment about the \\(x\\)-axis:} \\quad & M_x = \\int \\tilde{y} \\, dm \\\\[6pt]
\n\\text{Moment about the \\(y\\)-axis:} \\quad & M_y = \\int \\tilde{x} \\, dm \\\\[6pt]
\n\\text{Mass:} \\quad & M = \\int \\, dm \\\\[6pt]
\n\\text{Center of mass:} \\quad & \\overline{x} = \\frac{M_y}{M} ,\\,\\, \\overline{y} = \\frac{M_x}{M}
\n\\end{align}\\]\n<\/p><\/div>\n

\n

\uc608\uc81c 1.<\/span>
\n\uc88c\ud45c\ud3c9\uba74\uc5d0\uc11c \ud3ec\ubb3c\uc120 \\(y=4-x^2\\)\uacfc \\(x\\)\ucd95\uc73c\ub85c \ub458\ub7ec\uc2f8\uc778 \uc601\uc5ed\uc5d0 \ub193\uc778 \uc587\uc740 \ud310\uc758 \uc9c8\ub7c9\uc911\uc2ec\uc744 \uad6c\ud558\uc2dc\uc624. \ub2e8, \\((x,\\,y)\\)\uc5d0\uc11c\uc758 \ubc00\ub3c4\ub294 \\(\\delta = 2x^2\\)\uc774\ub2e4.\n<\/p>\n

\ud480\uc774.<\/span>
\n\uad6c\ud558\uace0\uc790 \ud558\ub294 \uc9c8\ub7c9\uc911\uc2ec\uc758 \uc88c\ud45c\ub97c \\((\\overline{x} ,\\, \\overline{y} )\\)\ub77c\uace0 \ud558\uc790. \uc77c\ub2e8 \uc8fc\uc5b4\uc9c4 \uc601\uc5ed\uc758 \ubaa8\uc591\uacfc \ubc00\ub3c4\ud568\uc218\uac00 \ubaa8\ub450 \\(y\\)\ucd95\uc5d0 \ub300\uce6d\uc774\ubbc0\ub85c \\(\\overline{x} =0 \\)\uc774\ub2e4. \uc774\uc81c \\(\\overline{y}\\)\ub9cc \uad6c\ud558\uba74 \ub41c\ub2e4.<\/p>\n

\uc8fc\uc5b4\uc9c4 \uc601\uc5ed\uc744 \\(y\\)\ucd95\uc5d0 \ud3c9\ud589\ud55c \uc9c1\uc120\uc73c\ub85c \uc798\ub77c \ub760\ub97c \ub9cc\ub4e4\uc790. \uadf8 \ubaa8\uc591\uc740 \uc544\ub798 \uadf8\ub9bc\uacfc \uac19\ub2e4.<\/p>\n

\"\"<\/p>\n

\ud558\ub098\uc758 \ub760\ub294 \uac00\ub85c\uc758 \uae38\uc774 \\(dx\\)\uac00 \\(0\\)\uc5d0 \uac00\uae4c\uc6b4 \ubaa8\uc591\uc774\ub77c\uace0 \uc0dd\uac01\ud558\uba74 \ub41c\ub2e4. \uac01 \ub760\uc758 \uc9c8\ub7c9\uc911\uc2ec\uc744 \\((\\tilde{x} ,\\, \\tilde{y} )\\)\ub77c\uace0 \ud558\uc790. \ub760\uc758 \uac00\ub85c\uc758 \uae38\uc774\uac00 \\(0\\)\uc5d0 \uac00\uae5d\uace0 \ubc00\ub3c4\ud568\uc218\ub294 \uc5f0\uc18d\ud568\uc218\uc774\ubbc0\ub85c \\(\\tilde{x} = x\\)\uc778 \uac83\uc73c\ub85c \ub450\uc5b4\ub3c4 \ubb34\ubc29\ud558\ub2e4. \ud55c\ud3b8 \ubc00\ub3c4\ud568\uc218\ub294 \\(y\\)\uc88c\ud45c\uc5d0 \uc601\ud5a5\uc744 \ubc1b\uc9c0 \uc54a\uc73c\ubbc0\ub85c \ub760\uc758 \uc138\ub85c\uc758 \ubc18\uc5d0 \ud574\ub2f9\ud558\ub294 \ubd80\ubd84\uc774 \\(\\tilde{y}\\)\uac00 \ub41c\ub2e4. \uc989
\n\\[\\tilde{x} = x ,\\,\\,\\, \\tilde{y} = \\frac{4-x^2}{2}\\]
\n\uc774\ub2e4. \\(\\overline{y}\\)\ub97c \uad6c\ud558\ub824\uba74 \\(x\\)\ucd95\uc5d0 \ub300\ud55c \ubaa8\uba58\ud2b8\ub97c \uacc4\uc0b0\ud574\uc57c \ud55c\ub2e4. \uba3c\uc800 \uc774 \ub760\uc758 \uc9c8\ub7c9\uc740
\n\\[dm = dx \\times (4-x^2 ) \\times \\delta = 2x^2 (4-x^2 ) dx \\]
\n\uc774\ub2e4. \ub610\ud55c \uc774 \ub760\uc758 \uc9c8\ub7c9\uc911\uc2ec\uc774 \\(x\\)\ucd95\uc73c\ub85c\ubd80\ud130 \ub5a8\uc5b4\uc9c4 \uac70\ub9ac\ub294 \\(\\tilde{y}\\)\uc774\ub2e4. \ub530\ub77c\uc11c
\n\\[\\begin{align}
\nM_x &= \\int \\tilde{y} dm \\\\[4pt]
\n&= \\int_{-2}^{2} \\frac{4-x^2}{2} \\cdot 2x^2 \\cdot (4-x^2 ) dx \\\\[4pt]
\n&= \\int_{-2}^{2} (16x^2 – 8x^4 + x^6 ) dx = \\frac{2048}{105}
\n\\end{align}\\]
\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
\n\\[\\overline{y} = \\frac{M_x}{M} = \\frac{2048}{105} \\cdot \\frac{15}{256} = \\frac{8}{7}\\]
\n\uc774\ub2e4.<\/p>\n

(Thomas\u2019 Calculus Global Edition 13\ud310 6.6\uc808\uc5d0\uc11c \ubc1c\ucdcc\ud568.)
\n<\/span><\/p>\n<\/div>\n

(4) \ud3c9\uba74\uc5d0\uc11c \ub450 \uace1\uc120\uc73c\ub85c \ub458\ub7ec\uc2f8\uc778 \uc601\uc5ed<\/h4>\n

\uc88c\ud45c\ud3c9\uba74\uc5d0 \ub193\uc778 \ud3c9\ud3c9\ud55c \ud310\uc774 \ub450 \ud568\uc218 \\(y=f(x)\\)\uc640 \\(y=g(x)\\)\uc758 \uadf8\ub798\ud504, \uadf8\ub9ac\uace0 \ub450 \uc9c1\uc120 \\(x=a,\\) \\(x=b\\)\ub85c \ub458\ub7ec\uc2f8\uc778 \ubaa8\uc591\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \ud3c9\uba74 \uc704\uc758 \uac01 \uc810\uc5d0\uc11c \ubc00\ub3c4\uac00 \\(\\delta\\)\ub77c\uace0 \ud558\uc790. [\ub2e8, \\(\\delta\\)\ub294 \\(y\\)\uc88c\ud45c\uc5d0 \uc601\ud5a5\uc744 \ubc1b\uc9c0 \uc54a\uace0 \uc624\uc9c1 \\(x\\)\ub9cc\uc744 \ubcc0\uc218\ub85c \ucde8\ud558\ub294 \ud568\uc218\ub77c\uace0 \ud558\uc790.] \uc774 \ud310\uc758 \uc9c8\ub7c9\uc911\uc2ec\uc744 \uad6c\ud558\uae30 \uc704\ud558\uc5ec, \uc544\ub798 \uadf8\ub9bc\uacfc \uac19\uc774 \\(y\\)\ucd95\uc5d0 \ud3c9\ud589\ud55c \uc9c1\uc120\uc73c\ub85c \uc798\ub77c \ub760\ub97c \ub9cc\ub4e4\uc790.<\/p>\n

\"\"<\/p>\n

\uc774 \ub760\uc758 \uc9c8\ub7c9\uc911\uc2ec\uc740
\n\\[(\\tilde{x} ,\\,\\tilde{y}) = \\left( x ,\\, \\frac{1}{2} [f(x)+g(x)] \\right)\\]
\n\uc774\ub2e4. \uc774 \ub760\uc758 \uac00\ub85c \uae38\uc774\ub294 \\(dx\\)\uc774\uba70 \uae38\uc774(\ub192\uc774)\ub294 \\((f(x)-g(x))\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc774 \ub760\uc758 \ub113\uc774\ub294
\n\\[dA = [f(x)-g(x)]dx\\]
\n\uc774\uba70, \uc774 \ub760\uc758 \uc9c8\ub7c9\uc740
\n\\[dm = \\delta \\,dA = \\delta [ f(x)-g(x)]dx\\]
\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(y\\)\ucd95\uc5d0 \ub300\ud55c \ud310\uc758 \ubaa8\uba58\ud2b8\ub294
\n\\[M_y = \\int x\\,dm = \\int_a^b x \\delta [f(x)-g(x)]dx\\]
\n\uc774\uba70, \\(x\\)\ucd95\uc5d0 \ub300\ud55c \ud310\uc758 \ubaa8\uba58\ud2b8\ub294
\n\\[\\begin{align}
\nM_x &= \\int y\\, dm \\\\[4pt]
\n&= \\int_a^b \\frac{1}{2} [f(x)+g(x)] \\cdot \\delta [f(x)-g(x)]dx\\\\[4pt]
\n&= \\int_a^b \\frac{\\delta}{2} [(f(x))^2 – (g(x))^2 ] dx
\n\\end{align}\\]
\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ud310 \uc804\uccb4\uc758 \uc9c8\ub7c9\uc911\uc2ec \\((\\overline{x} ,\\, \\overline{y} )\\)\uc758 \uc88c\ud45c\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n

\n

\uc815\ub9ac 1. (\ub450 \ud568\uc218\uc758 \uadf8\ub798\ud504\ub85c \ub458\ub7ec\uc2f8\uc778 \ud310 \ubaa8\uc591\uc758 \uc9c8\ub7c9\uc911\uc2ec)<\/span><\/p>\n

\\[\\begin{align}
\n\\overline{x} &= \\frac{1}{M} \\int_a ^b \\delta x [f(x)-g(x)] dx, \\\\[4pt]
\n\\overline{y} &= \\frac{1}{M} \\int_a ^b \\delta [(f(x))^2 – (g(x))^2 ]dx.
\n\\end{align}\\]\n<\/p><\/div>\n

\ubc00\ub3c4\ud568\uc218\uac00 \uc0c1\uc218\uc77c \ub54c \ubb3c\uccb4\uc758 \uc9c8\ub7c9\uc911\uc2ec\uc744 \uadf8 \ubb3c\uccb4\uc758 \ubaa8\uc591\uc911\uc2ec<\/span>(centroid) \ub610\ub294 \ubb34\uac8c\uc911\uc2ec<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\n

\uc608\uc81c 2.<\/span>
\n\ud3c9\uba74\uc5d0\uc11c \ud3ec\ubb3c\uc120 \\(y=x^2\\)\uacfc \uc9c1\uc120 \\(y=x\\)\ub85c \ub458\ub7ec\uc2f8\uc778 \uc601\uc5ed\uc758 \ubaa8\uc591\uc911\uc2ec\uc744 \uad6c\ud558\uc2dc\uc624.\n<\/p>\n

\ud480\uc774.<\/span>
\n\\(\\delta = 1\\)\ub85c \ub450\uace0 \uc815\ub9ac 1\uc758 \uacf5\uc2dd\uc744 \uc774\uc6a9\ud558\uc790. \uba3c\uc800 \uc601\uc5ed\uc758 \ub113\uc774\ub294
\n\\[M = \\int_0^1 (x-x^2 )dx = \\frac{1}{6}\\]
\n\uc774\uba70, \ubaa8\uba58\ud2b8\ub294
\n\\[\\begin{align}
\nM_x &= \\int_0^1 \\frac{1}{2}(x+x^2)(x-x^2) dx = \\frac{1}{15} , \\\\[6pt]
\nM_y &= \\int_0^1 x(x-x^2)dx = \\frac{1}{12}
\n\\end{align}\\]
\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc601\uc5ed\uc758 \ubaa8\uc591\uc911\uc2ec \\((\\overline{x} ,\\, \\overline{y} )\\)\uc758 \uc88c\ud45c\ub294
\n\\[\\overline{x} = \\frac{M_y}{M} = \\frac{1}{2} ,\\,\\,\\,
\n\\overline{y} = \\frac{M_x}{M} = \\frac{2}{5}\\]
\n\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n

(5) \ud3c9\uba74\uc5d0 \ub193\uc778 \uace1\uc120<\/h4>\n

\uace1\uc120\uc73c\ub85c \ub458\ub7ec\uc2f8\uc778 \uc601\uc5ed\uc758 \uc9c8\ub7c9\uc911\uc2ec\ubfd0\ub9cc \uc544\ub2c8\ub77c \uace1\uc120\uc758 \uc9c8\ub7c9\uc911\uc2ec\ub3c4 \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\n

\n

\uc608\uc81c 3.<\/span>
\n\uc911\uc2ec\uc774 \uc6d0\uc810\uc774\uace0 \ubc18\uc9c0\ub984\uc774 \\(a > 0\\)\uc774\uba70 \uc0c1\ubc18\ud3c9\uba74\uc5d0 \ub193\uc778 \ubc18\uc6d0 \\(y=\\sqrt{a^2 – x^2}\\)\uc758 \ubaa8\uc591\uc911\uc2ec\uc744 \uad6c\ud558\uc2dc\uc624.\n<\/p>\n

\ud480\uc774.<\/span>
\n\ubc00\ub3c4\ud568\uc218\ub97c \\(\\delta = 1\\)\uc774\ub77c\uace0 \ub450\uc790. \ubc18\uc6d0\uc758 \ubaa8\uc591\uc911\uc2ec\uc758 \uc88c\ud45c\ub97c \\((\\overline{x} ,\\, \\overline{y})\\)\ub77c\uace0 \ud558\uc790. \ubc18\uc6d0\uc774 \\(y\\)\ucd95\uc5d0 \ub300\uce6d\uc774\ubbc0\ub85c \\(\\overline{x} =0\\)\uc774\ub2e4. \uc774\uc81c \\(\\overline{y}\\)\ub9cc \uad6c\ud558\uba74 \ub41c\ub2e4.<\/p>\n

\ubc18\uc6d0\uc744 \uc791\uc740 \ud638(arc) \uc870\uac01\uc73c\ub85c \ub098\ub204\uc790. \ud558\ub098\uc758 \ud638 \uc870\uac01\uc758 \uc9c8\ub7c9\uc911\uc2ec\uc744 \\((\\tilde{x} ,\\, \\tilde{y})\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(\\tilde{y} = a\\sin\\theta\\)\ub77c\uace0 \ub098\ud0c0\ub0bc \uc218 \uc788\uc73c\uba70, \ud638 \uc870\uac01\uc758 \uae38\uc774\ub294 \\(ds = a d\\theta\\)\ub77c\uace0 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4. \uc774 \ud638 \uc870\uac01\uc758 \uc9c8\ub7c9\uc740\\[dm = \\delta \\,ds = a\\,d\\theta\\]\uc774\ub2e4. \ub610\ud55c \ud638 \uc870\uac01\uc758 \uc9c8\ub7c9\uc911\uc2ec\uc774 \\(x\\)\ucd95\uc73c\ub85c\ubd80\ud130 \ub5a8\uc5b4\uc9c4 \uac70\ub9ac\ub294\\[\\tilde{y} = a\\sin \\theta\\]\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
\n\\[
\n\\overline{y} = \\frac{\\int \\tilde{y} \\,dm}{\\int dm}
\n= \\frac{\\int_0^{\\pi} a \\sin \\theta \\cdot a \\,d\\theta}{\\int_0^{\\pi} a\\,d\\theta}
\n= \\frac{2a^2}{a\\pi} = \\frac{2a}{\\pi}
\n\\]
\n\uc774\ub2e4.
\n(Thomas\u2019 Calculus Global Edition 13\ud310 6.6\uc808\uc5d0\uc11c \ubc1c\ucdcc\ud568.)
\n<\/span><\/p>\n<\/div>\n

<\/p>\n

\ud68c\uc804\uccb4\uc758 \ubd80\ud53c\uc640 \ub113\uc774\uc5d0 \ub300\ud55c \ud30c\ud478\uc2a4\uc758 \uc815\ub9ac<\/h3>\n

4\uc138\uae30 \uacbd \uc54c\ub809\uc0b0\ub4dc\ub9ac\uc544\uc758 \uc218\ud559\uc790 \ud30c\ud478\uc2a4\ub294 \ub3c4\ud615\uc758 \ubaa8\uc591\uc911\uc2ec\uc744 \uc774\uc6a9\ud558\uc5ec \ud68c\uc804\uccb4\uc758 \ubd80\ud53c\uc640 \ub113\uc774\ub97c \uad6c\ud558\ub294 \ubc29\ubc95\uc744 \uace0\uc548\ud558\uc600\ub2e4.<\/p>\n

\n

\uc815\ub9ac 2. (\ud68c\uc804\uccb4\uc758 \ubd80\ud53c\uc5d0 \ub300\ud55c \ud30c\ud478\uc2a4\uc758 \uc815\ub9ac)<\/span><\/p>\n

\ud3c9\uba74\uc5d0 \ub113\uc774\uac00 \\(A\\)\uc778 \ub3c4\ud615\uc774 \uc788\ub2e4. \uc774 \ub3c4\ud615\uc774 \ub3c4\ud615\uc744 \uad00\ud1b5\ud558\uc9c0 \uc54a\uace0 \ub3d9\uc77c\ud3c9\uba74\uc5d0 \ub193\uc778 \uc9c1\uc120 \\(\\ell\\)\uc744 \ucd95\uc73c\ub85c \ud68c\uc804\ud558\uc5ec \ub9cc\ub4e0 \ud68c\uc804\uccb4\uc758 \ubd80\ud53c\ub294 \ub3c4\ud615\uc758 \ubaa8\uc591\uc911\uc2ec\uc774 \uc9c1\uc120 \\(\\ell\\)\uc744 \ucd95\uc73c\ub85c \ud68c\uc804\ud558\uc600\uc744 \ub54c\uc758 \uc790\ucde8(\uc6d0)\uc758 \uae38\uc774\uc5d0 \ub3c4\ud615\uc758 \ub113\uc774\ub97c \uacf1\ud55c \uac83\uacfc \uac19\ub2e4. \uc989 \ub3c4\ud615\uc758 \ubaa8\uc591\uc911\uc2ec\uacfc \uc9c1\uc120 \\(\\ell\\)\uc758 \uac70\ub9ac\ub97c \\(\\rho\\)\ub77c\uace0 \ud558\uba74 \ud68c\uc804\uccb4\uc758 \ubd80\ud53c\ub294
\n\\[V = 2\\pi \\rho A\\]
\n\uc774\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/p>\n

\ub3c4\ud615\uc774 \uc5f0\uc18d\uc778 \uace1\uc120\uc73c\ub85c \ub458\ub7ec\uc2f8\uc5ec \uc788\uace0 \ubcfc\ub85d\ud55c \uacbd\uc6b0\uc5d0 \ub300\ud558\uc5ec \uc99d\uba85\ud55c\ub2e4.
\n\ub3c4\ud615\uc774 \\(xy\\)-\ud3c9\uba74\uc758 \uc81c 1 \uc0ac\ubd84\uba74\uc5d0 \ub193\uc5ec \uc788\uace0 \\(x\\)\ucd95\uc744 \ucd95\uc73c\ub85c \ud68c\uc804\ud55c\ub2e4\uace0 \ud558\uc790. \uc8fc\uc5b4\uc9c4 \ub3c4\ud615\uc744 \\(x\\)\ucd95\uc5d0 \ud3c9\ud589\ud55c \uc9c1\uc120\uc73c\ub85c \uc798\ub77c \ub760\ub97c \ub9cc\ub4e4\uc790. \ub760\uac00 \\(x\\)\ucd95\uc73c\ub85c\ubd80\ud130 \ub5a8\uc5b4\uc9c4 \uac70\ub9ac\ub97c \\(\\tilde{y} = y\\)\ub77c\uace0 \ud558\uace0, \ub760\uc758 \uae38\uc774\ub97c \\(L(y)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(L(y)\\)\ub294 \\(y\\)\ub97c \ubcc0\uc218\ub85c \ud558\ub294 \uc5f0\uc18d\ud568\uc218\uc774\ub2e4.<\/p>\n

\ub3c4\ud615\uc774 \ucc28\uc9c0\ud558\ub294 \uc601\uc5ed\uc758 \\(y\\)\uc88c\ud45c\uc758 \ubc94\uc704\uac00 \\(c \\le y \\le d\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc774 \ub3c4\ud615\uc744 \\(x\\)\ucd95\uc5d0 \ub300\ud558\uc5ec \ud68c\uc804\ud55c \ud68c\uc804\uccb4\uc758 \ubd80\ud53c\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.
\n\\[\\begin{align}
\nV &= \\int_c^d 2 \\pi \\text{(shell radius)(shell height)}dy \\\\[6pt]
\n&= 2 \\pi \\int_c^d y\\,L(y)\\,dy.
\n\\end{align}\\]
\n\uadf8\ub7ec\ubbc0\ub85c \ub3c4\ud615\uc758 \ubaa8\uc591\uc911\uc2ec\uc758 \\(y\\)\uc88c\ud45c\ub294
\n\\[\\overline{y} = \\frac{1}{A} \\int_c^d \\tilde{y} dA = \\frac{1}{A} \\int_c^d y \\,L(y) \\,dy\\]
\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
\n\\[V = 2\\pi \\int_c^d y\\,L(y) \\,dy = 2\\pi A\\overline{y} = 2\\pi \\rho A\\]
\n\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\n

\uc815\ub9ac 3. (\ud68c\uc804\uccb4\uc758 \ub113\uc774\uc5d0 \ub300\ud55c \ud30c\ud478\uc2a4\uc758 \uc815\ub9ac)<\/span><\/p>\n

\ud3c9\uba74\uc5d0 \uae38\uc774\uac00 \\(L\\)\uc778 \uace1\uc120\uc774 \uc788\ub2e4. \uc774 \uace1\uc120\uc774 \ub3d9\uc77c\ud3c9\uba74\uc5d0 \ub193\uc5ec \uc788\uace0 \uace1\uc120\uc744 \uc9c0\ub098\uc9c0 \uc54a\ub294 \uc9c1\uc120 \\(\\ell\\)\uc744 \ucd95\uc73c\ub85c \ud68c\uc804\ud558\uc5ec \ub9cc\ub4e0 \ud68c\uc804\uccb4\uc758 \ub113\uc774\ub294 \uace1\uc120\uc758 \ubaa8\uc591\uc911\uc2ec\uc774 \uc9c1\uc120 \\(\\ell\\)\uc744 \ucd95\uc73c\ub85c \ud68c\uc804\ud558\uc600\uc744 \ub54c\uc758 \uc790\ucde8(\uc6d0)\uc758 \uae38\uc774\uc5d0 \uace1\uc120\uc758 \uae38\uc774\ub97c \uacf1\ud55c \uac83\uacfc \uac19\ub2e4. \uc989 \uace1\uc120\uc758 \ubaa8\uc591\uc911\uc2ec\uacfc \uc9c1\uc120 \\(\\ell\\)\uc758 \uac70\ub9ac\ub97c \\(\\rho\\)\ub77c\uace0 \ud558\uba74 \ud68c\uc804\uccb4\uc758 \ub113\uc774\ub294
\n\\[S = 2\\pi \\rho L\\]
\n\uc774\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/p>\n

\uace1\uc120\uc774 \ub9e4\ub044\ub7ec\uc6b4 \uacbd\uc6b0(\ubbf8\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uc758 \uadf8\ub798\ud504\uc778 \uacbd\uc6b0)\uc5d0 \ub300\ud558\uc5ec \uc99d\uba85\ud55c\ub2e4.
\n\uace1\uc120\uc774 \\(xy\\)-\ud3c9\uba74\uc758 \uc81c 1 \uc0ac\ubd84\uba74\uc5d0 \ub193\uc5ec \uc788\uace0, \uc88c\ud45c\ud3c9\uba74\uc5d0\uc11c \uc774 \uace1\uc120\uc774 \ucc28\uc9c0\ud558\ub294 \uc601\uc5ed\uc758 \\(x\\)\uc88c\ud45c\uc758 \ubc94\uc704\uac00 \\(a \\le x \\le b\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uc774 \uace1\uc120\uc774 \\(x\\)\ucd95\uc744 \ucd95\uc73c\ub85c \ud68c\uc804\ud55c\ub2e4\uace0 \ud558\uc790.<\/p>\n

\uace1\uc120\uc744 \uc5ec\ub7ec \uac1c\uc758 \ud638 \uc870\uac01\uc73c\ub85c \ub098\ub208\ub2e4. \uadf8 \uc911 \ud55c \uc870\uac01\uc744 \uc0dd\uac01\ud558\uc790. \ud638 \uc870\uac01\uc758 \uae38\uc774\ub97c \\(ds\\)\ub77c\uace0 \ud558\uace0, \ud638 \uc870\uac01\uc774 \\(x\\)\ucd95\uc73c\ub85c\ubd80\ud130 \ub5a8\uc5b4\uc9c4 \uac70\ub9ac\ub97c \\(\\tilde{y} = y\\)\ub77c\uace0 \ud558\uc790. \uace1\uc120\uc774 \ub9e4\ub044\ub7fd\uace0 \ud638 \uc870\uac01\uc758 \uae38\uc774\uac00 \ub9e4\uc6b0 \uc9e7\uc73c\ubbc0\ub85c, \uc774 \ud638 \uc870\uac01\uc774 \ud68c\uc804\ud55c \uc790\ucde8\uc758 \ub113\uc774\ub294 \\(2\\pi y \\,ds\\)\uc640 \uac19\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uace1\uc120\uc744 \ud68c\uc804\ud55c \ud68c\uc804\uccb4\uc758 \ub113\uc774\ub294
\n\\[S = \\int_{x=a}^{y=b} 2\\pi y\\,ds = 2\\pi \\int_{x=a}^{x=b} y\\,ds\\]
\n\uc774\ub2e4. \ud55c\ud3b8 \uace1\uc120\uc758 \ubaa8\uc591\uc911\uc2ec\uc758 \\(y\\)\uc88c\ud45c\ub294
\n\\[\\overline{y} = \\frac{\\int_{x=a}^{x=b} \\tilde{y} \\,ds}{\\int_{x=a}^{x=b} ds} = \\frac{1}{L} \\int_{x=a}^{x=b} y\\,ds\\]
\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c
\n\\[S = 2\\pi \\int_{x=a}^{x=b} y\\,ds = 2\\pi \\overline{y} L = 2\\pi \\rho L\\]
\n\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n

4\uc138\uae30\ub294 \ubbf8\uc801\ubd84\uc758 \uac1c\ub150\uc774 \uc815\ub9bd\ub418\uae30 \ud55c\ucc38 \uc804\uc778\ub370, \uc774\ub7ec\ud55c \ubc29\ubc95\uc744 \uace0\uc548\ud574\ub0c8\ub2e4\ub294 \uac83\uc740 \ucc38 \ub300\ub2e8\ud55c \uc77c\uc774\ub2e4.<\/p>\n

\n

\uc608\uc81c 4.<\/span>
\n\uc88c\ud45c\ud3c9\uba74\uc5d0\uc11c \uc911\uc2ec\uc774 \\((b,\\,0)\\)\uc774\uace0 \ubc18\uc9c0\ub984\uc758 \uae38\uc774\uac00 \\(a\\)\uc778 \uc6d0\uc744 \\(y\\)\ucd95\uc744 \ucd95\uc73c\ub85c \ud68c\uc804\uc2dc\ud0a8 \ud68c\uc804\uccb4(\uc6d0\ud658\uba74)\uc758 \ubd80\ud53c\uc640 \uac89\ub113\uc774\ub97c \uad6c\ud558\uc2dc\uc624. (\ub2e8, \\(b > a > 0.\\))\n<\/p>\n

\"\"<\/p>\n

\ud480\uc774.<\/span>
\n\uc8fc\uc5b4\uc9c4 \uc6d0\uc740 \uc911\uc2ec\uc774 \uace7 \ubaa8\uc591\uc911\uc2ec\uc774\ubbc0\ub85c, \ubaa8\uc591\uc911\uc2ec\uacfc \ud68c\uc804\ucd95\uc758 \uac70\ub9ac\ub294 \\(b\\)\uc774\ub2e4. \ub610\ud55c \uc774 \uc6d0\uc758 \ub113\uc774\ub294 \\(\\pi a^2\\)\uc774\uba70, \uc6d0\uc758 \ub458\ub808\uc758 \uae38\uc774\ub294 \\(2\\pi a\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ubb38\uc81c\uc758 \uc6d0\ud658\uba74\uc758 \ubd80\ud53c\ub294
\n\\[V = 2\\pi b \\times \\pi a^2 = 2 \\pi^2 a^2 b\\]
\n\uc774\uba70, \uc6d0\ud658\uba74\uc758 \uac89\ub113\uc774\ub294
\n\\[S = 2\\pi b \\times 2\\pi a = 4\\pi^2 ab\\]
\n\uc774\ub2e4.
\n(Thomas\u2019 Calculus Global Edition 13\ud310 6.6\uc808\uc758 \uadf8\ub9bc\uc744 \uc0ac\uc6a9\ud568.)
\n<\/span><\/p>\n<\/div>\n

\n

\uc608\uc81c 5.<\/span>
\n\\(xy\\)-\ud3c9\uba74\uc5d0\uc11c \ud3ec\ubb3c\uc120 \\(y=x^2\\)\uacfc \uc9c1\uc120 \\(y=x\\)\ub85c \ub458\ub7ec\uc2f8\uc778 \uc601\uc5ed \\(R\\)\ub97c \uc9c1\uc120 \\(y=x\\)\ub97c \ucd95\uc73c\ub85c \ud68c\uc804\ud55c \ud68c\uc804\uccb4\uc758 \ubd80\ud53c\ub97c \uad6c\ud558\uc2dc\uc624.\n<\/p>\n

\ud480\uc774.<\/span>
\n\uc608\uc81c 2\uc5d0\uc11c \\(R\\)\uc758 \ubaa8\uc591\uc911\uc2ec\uc740 \\((\\overline{x} ,\\, \\overline{y}) = (\\frac{1}{2} ,\\, \\frac{2}{5})\\)\uc774\uba70, \\(R\\)\uc758 \ub113\uc774\ub294 \\(\\frac{1}{6}\\)\uc774\ub2e4. \ubaa8\uc591\uc911\uc2ec\uc774 \uc9c1\uc120 \\(y=x\\)\uc640 \ub5a8\uc5b4\uc9c4 \uac70\ub9ac\ub294, \uc810 \\(\\left(\\frac{1}{2}, \\, \\frac{2}{5}\\right)\\)\uc640 \uc9c1\uc120 \\(x-y=0\\) \uc0ac\uc774\uc758 \uac70\ub9ac \uacf5\uc2dd\uc5d0 \uc758\ud558\uc5ec
\n\\[\\rho = \\frac{\\left\\lvert \\frac{1}{2} – \\frac{2}{5} \\right\\rvert}{\\sqrt{2}} = \\frac{1}{10\\sqrt{2}}\\]
\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ud68c\uc804\uccb4\uc758 \ubd80\ud53c\ub294
\n\\[V = 2\\pi \\times \\frac{1}{10\\sqrt{2}} \\times \\frac{1}{6} = \\frac{\\sqrt{2}}{60}\\pi\\]
\n\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n

\n

\uc608\uc81c 6.<\/span>
\n\ub124 \uc810 \\(\\mathrm{A}(0,\\,2),\\) \\(\\mathrm{B}(2,\\,0),\\) \\(\\mathrm{C}(4,\\,2),\\) \\(\\mathrm{D}(2,\\,4)\\)\ub97c \uaf2d\uc9d3\uc810\uc73c\ub85c \ud558\ub294 \uc0ac\uac01\ud615 \\(\\mathrm{ABCD}\\)\ub97c \uc9c1\uc120 \\(\\sqrt{3} (y+2) = x-2\\)\ub97c \ucd95\uc73c\ub85c \ud68c\uc804\ud558\uc5ec \uc5bb\uc740 \ud68c\uc804\uccb4\uc758 \uac89\ub113\uc774\ub97c \uad6c\ud558\uc2dc\uc624.\n<\/p>\n

\ud480\uc774.<\/span>
\n\uc0ac\uac01\ud615 \\(\\mathrm{ABCD}\\)\uc740 \uc815\uc0ac\uac01\ud615\uc774\ubbc0\ub85c \uc774 \uc0ac\uac01\ud615\uc758 \ubaa8\uc591\uc911\uc2ec\uc740 \ub450 \ub300\uac01\uc120\uc758 \uad50\uc810 \\((2,\\,2)\\)\uc774\ub2e4. \uc774 \uc810\uc774 \ud68c\uc804\ucd95 \\[x-\\sqrt{3} y – 2-2\\sqrt{3} =0\\]\uacfc \ub5a8\uc5b4\uc9c4 \uac70\ub9ac\ub294\\[\\rho = \\frac{\\left\\lvert 2-2\\sqrt{3} -2-2\\sqrt{3}\\right\\rvert}{\\sqrt{1+3}} = 2\\sqrt{3}\\]\uc774\ub2e4. \ud55c\ud3b8 \uc0ac\uac01\ud615 \\(\\mathrm{ABCD}\\)\uc758 \ub458\ub808 \uae38\uc774\ub294 \\(8\\sqrt{2}\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \ud68c\uc804\uccb4\uc758 \uac89\ub113\uc774\ub294
\n\\[S = 2\\pi \\times 2\\sqrt{3} \\times 8\\sqrt{2} = 32\\sqrt{6} \\pi\\]
\n\uc774\ub2e4.
\n<\/span><\/p>\n<\/div>\n

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\ucca8\uc5b8<\/h3>\n

\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ud3c9\uba74\uc5d0 \ub193\uc778 \ubb3c\uccb4\uc758 \uc9c8\ub7c9\uc911\uc2ec\uc744 \uad6c\ud560 \ub54c \ubc00\ub3c4\ud568\uc218 \\(\\delta\\)\uac00 \\(y\\)\uc88c\ud45c\uc5d0 \uc601\ud5a5\uc744 \ubc1b\uc9c0 \uc54a\uace0 \\(x\\)\ub9cc\uc744 \ubcc0\uc218\ub85c \ucde8\ud558\ub294 \ud568\uc218\uc778 \uacbd\uc6b0\ub9cc\uc744 \ub2e4\ub8e8\uc5c8\ub2e4. \\(\\delta\\)\uac00 \\(x\\)\uc88c\ud45c\uc640 \\(y\\)\uc88c\ud45c \ubaa8\ub450\uc5d0 \uc601\ud5a5\uc744 \ubc1b\ub294 2\ubcc0\uc218 \ud568\uc218\uc77c \ub54c \uc9c8\ub7c9\uc911\uc2ec\uc744 \uad6c\ud558\ub824\uba74 \uc911\uc801\ubd84\uc744 \uc0ac\uc6a9\ud574\uc57c \ud55c\ub2e4.<\/p>\n

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<\/p>\n","protected":false},"excerpt":{"rendered":"

\uc774 \ud3ec\uc2a4\ud2b8\uc5d0\uc11c\ub294 \ubaa8\uba58\ud2b8\uc640 \uc9c8\ub7c9\uc911\uc2ec\uc744 \uc218\ud559\uc801\uc73c\ub85c \uc815\uc758\ud558\uace0, \uc9c8\ub7c9\uc911\uc2ec\uc744 \uacc4\uc0b0\ud558\ub294 \uacf5\uc2dd\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \ub354\ubd88\uc5b4 \uc9c8\ub7c9\uc911\uc2ec\uc744 \uc774\uc6a9\ud558\uc5ec \ud68c\uc804\uccb4\uc758 \ubd80\ud53c\uc640 \ub113\uc774\ub97c \uc27d\uac8c \uacc4\uc0b0\ud560 \uc218 \uc788\ub294 \ud30c\ud478\uc2a4\uc758 \uc815\ub9ac\ub97c \uc0b4\ud3b4\ubcf8\ub2e4. \ubaa8\uba58\ud2b8\uc640 \uc9c8\ub7c9\uc911\uc2ec\uc758 \uc815\uc758 \ubaa8\uba58\ud2b8\uc640 \uc9c8\ub7c9\uc911\uc2ec\uc740 \uc138 \ub2e8\uacc4\ub85c \uc815\uc758\ud55c\ub2e4. \uba3c\uc800 \uc9c1\uc120 \uc704\uc5d0 \ub193\uc778 \uc720\ud55c \uac1c\uc758 \ubb3c\uccb4\uc5d0 \ub300\ud558\uc5ec \uc815\uc758\ud558\uace0, \ub2e4\uc74c\uc73c\ub85c \ud3c9\uba74\uc5d0 \ub193\uc778 \uc720\ud55c \uac1c\uc758 \ubb3c\uccb4\uc5d0 \ub300\ud558\uc5ec \uc815\uc758\ud55c \ub4a4, \ub9c8\uc9c0\ub9c9\uc73c\ub85c \ud3c9\uba74\uc5d0 \ub193\uc778 \ubb3c\uccb4(\uac01 \uc88c\ud45c\uc5d0\uc11c \ubc00\ub3c4\uac00 \ud568\uc218\ub85c \uc8fc\uc5b4\uc9c4)\uc5d0 \ub300\ud558\uc5ec \uc815\uc758\ud55c\ub2e4. (1) \uc9c1\uc120 \uc704\uc5d0 \ub193\uc778 \uc720\ud55c…<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"categories":[47],"tags":[272,32,274,270,271,273],"class_list":["post-1993","post","type-post","status-publish","format-standard","hentry","category-calculus-ap","tag-center-of-mass","tag-moment","tag-pappus-theorem","tag-270","tag-271","tag-273"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1993","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/comments?post=1993"}],"version-history":[{"count":56,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1993\/revisions"}],"predecessor-version":[{"id":3299,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/posts\/1993\/revisions\/3299"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=1993"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/categories?post=1993"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/tags?post=1993"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}