{"id":6739,"date":"2021-07-21T00:10:54","date_gmt":"2021-07-20T15:10:54","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6739"},"modified":"2022-03-06T19:48:26","modified_gmt":"2022-03-06T10:48:26","slug":"length-area-and-volume","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/length-area-and-volume\/","title":{"rendered":"\uae38\uc774, \ub113\uc774, \ubd80\ud53c"},"content":{"rendered":"<div style=\"display: none; visibility: hidden;\">\n<style type=\"text\/css\">\n\timg.mfp-img { background-color: white; }\n<\/style>\n<p><!--\n\\[\n\\newcommand{\\vecf}{{\\mathbf{f}}}\n\\newcommand{\\vecL}{{\\mathbf{L}}}\n\\newcommand{\\vecR}{{\\mathbb{R}}}\n\\newcommand{\\imI}{\\boldsymbol{i}}\n\\]\n-->\n<\/div>\n<div class=\"box itc_intro\">\n<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 6\uc7a5 4\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; <span class=\"itc_viewcontents\">(<a href=\"..\/\">\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30<\/a>)<\/span><\/p>\n<\/div>\n<p>\uc774 \uc808\uc5d0\uc11c\ub294 \uc801\ubd84\uc744 \uc774\uc6a9\ud558\uc5ec \uace1\uc120\uc758 \uae38\uc774, \ud68c\uc804\uba74\uc758 \ub113\uc774, \uc785\uccb4\ub3c4\ud615\uc758 \ubd80\ud53c \uad6c\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uc790.<\/p>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uace1\uc120\uc758 \uae38\uc774<\/h2>\n<p>\ud568\uc218 \\(f\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub41c \ubbf8\ubd84 \uac00\ub2a5\ud55c \uc2e4\uc22b\uac12 \ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\uc81c \\(a\\le x le b\\)\uc758 \ubc94\uc704\uc5d0\uc11c \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504\uc758 \uae38\uc774\ub97c \uad6c\ud574 \ubcf4\uc790.<\/p>\n<p>\uad6c\uac04 \\([a,\\,b]\\)\ub97c \\(n\\)\uac1c\uc758 \uc18c\uad6c\uac04 \\(\\left[ x_{i-1},\\, x_i \\right]\\)\ub85c \ubd84\ud560\ud558\uc790. \uc989<br \/>\n\\[a = x_0 < x_1 < x_2 < \\cdots < x_n = b\\]\n\uc774\ub2e4. \\(k\\)\ubc88\uc9f8 \uc18c\uad6c\uac04 \\(\\left[ x_{k-1},\\, x_k \\right]\\)\uc758 \uae38\uc774\ub97c \\(\\Delta x_k\\)\ub77c\uace0 \ud558\uc790. \uac01 \uc810 \\(x_k\\)\ub294 \uace1\uc120 \uc704\uc758 \uc810 \\(\\mathrm{P}_k ( x_k ,\\, f(x_k ))\\)\uc5d0 \ub300\uc751\ub41c\ub2e4.<\/p>\n<div style=\"margin-top: 1.5em; margin-bottom: 1.5em\">\n<a href=\"\/blog\/wp-content\/uploads\/2021\/09\/ic_704_1a.png\" data-rel=\"penci-gallery-image-content\" ><img fetchpriority=\"high\" decoding=\"async\" src=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_1a.png\" alt=\"\" width=\"340\" height=\"223\" class=\"aligncenter size-full wp-image-7815\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_1a.png 1699w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_1a-300x197.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_1a-1024x673.png 1024w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_1a-768x505.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_1a-1536x1010.png 1536w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_1a-1170x769.png 1170w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_1a-585x385.png 585w\" sizes=\"(max-width: 340px) 100vw, 340px\" \/><\/a>\n<\/div>\n<p>\\(n\\)\uc758 \uac12\uc774 \ucee4\uc9c8 \uc218\ub85d \uc120\ubd84 \\(\\overline{\\mathrm{P}_{k-1}\\mathrm{P}_k}\\)\uc758 \uae38\uc774\uc758 \ud569<br \/>\n\\[\\overline{\\mathrm{P}_0\\mathrm{P}_1} + \\overline{\\mathrm{P}_1\\mathrm{P}_2} + \\cdots + \\overline{\\mathrm{P}_{n-1}\\mathrm{P}_n} = \\sum_{k=1}^{n} \\overline{\\mathrm{P}_{k-1}\\mathrm{P}_k}\\]<br \/>\n\uc740 \uace1\uc120\uc758 \uae38\uc774\uc5d0 \uac00\uae4c\uc6cc\uc9c4\ub2e4.<\/p>\n<div style=\"margin-top: 1.5em; margin-bottom: 1.5em\">\n<a href=\"\/blog\/wp-content\/uploads\/2021\/09\/ic_704_1b.png\" data-rel=\"penci-gallery-image-content\" ><img decoding=\"async\" src=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_1b.png\" alt=\"\" width=\"340\" height=\"223\" class=\"aligncenter size-full wp-image-7816\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_1b.png 1699w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_1b-300x197.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_1b-1024x673.png 1024w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_1b-768x505.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_1b-1536x1010.png 1536w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_1b-1170x769.png 1170w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_1b-585x385.png 585w\" sizes=\"(max-width: 340px) 100vw, 340px\" \/><\/a>\n<\/div>\n<p>\uc120\ubd84 \ud55c \uc870\uac01\uc758 \uae38\uc774\ub294<br \/>\n\\[\\overline{\\mathrm{P}_{k-1}\\mathrm{P}_k} = \\sqrt{ (x_k &#8211; x_{k-1})^2 + (f(x_k) &#8211; f(x_{k-1}))^2 }\\]<br \/>\n\uc774\ubbc0\ub85c \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec \\(x_k ^* \\in ( x_{k-1} ,\\, x_k )\\)\uac00 \uc874\uc7ac\ud558\uc5ec<br \/>\n\\[f(x_k ) &#8211; f(x_{k-1}) = f &#8216; (x_k ^* ) \\Delta x_k\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a8\ub2e4. \uc5ec\uae30\uc11c \\(\\Delta x_k &#8211; x_k &#8211; x_{k-1}\\)\uc774\ub2e4. \ub530\ub77c\uc11c<br \/>\n\\[\\begin{align}<br \/>\n\\overline{\\mathrm{P}_{k-1}\\mathrm{P}_k}<br \/>\n&#038;= \\sqrt{ (\\Delta x_k )^2 + (\\Delta x_k )^2 ( f &#8216; (x_k ^* ))^2 } \\\\[5pt]<br \/>\n&#038;= \\sqrt{ (1+ (f &#8216; (x_k ^* ))^2 ) (\\Delta x_k )^2} \\\\[5pt]<br \/>\n&#038;= \\sqrt{1+ (f &#8216; (x_k ^* ))^2} \\Delta x_k<br \/>\n\\end{align}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc120\ubd84 \uc870\uac01\ub4e4\uc758 \uae38\uc774\uc758 \ud569\uc740<br \/>\n\\[\\sum_{k=1}^n \\overline{\\mathrm{P}_{k-1}\\mathrm{P}_k}<br \/>\n=\\sum_{k=1}^n \\sqrt{ 1+(f &#8216; (x_k ^* ))^2 }\\Delta x_k\\]<br \/>\n\uc774\ub2e4. \uc774 \ub4f1\uc2dd\uc758 \uc6b0\ubcc0\uc740 \uc815\uc801\ubd84\uc758 \ubd80\ubd84\ud569\uacfc \uac19\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(x_k\\)\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ubd84\ud560\uc744 \\(P\\)\ub77c \ud560 \ub54c, \\(\\lVert P \\rVert \\rightarrow 0\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uba74 \uc704 \ud569\uc740<br \/>\n\\[\\int_a^b \\sqrt{ 1+ ( f &#8216; (x))^2 } \\,dx\\]<br \/>\n\uc5d0 \uc218\ub834\ud55c\ub2e4.<\/p>\n<p>\uc9c0\uae08\uae4c\uc9c0 \ub2e4\uc74c \uacf5\uc2dd\uc744 \uc720\ub3c4\ud558\uc600\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 6.4.1. (\uace1\uc120\uc758 \uae38\uc774)<\/span><\/p>\n<p>\\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud55c \uc2e4\uc22b\uac12 \ud568\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(a \\le x \\le b\\)\uc5d0\uc11c \ud568\uc218 \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504\uc758 \uae38\uc774 \\(L\\)\uc740<br \/>\n\\[L = \\int_a^b \\sqrt{1+ (f &#8216; (x))^2} \\,dx \\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 6.4.1.<\/span><\/p>\n<p>\\(0\\le x \\le 5\\)\uc758 \ubc94\uc704\uc5d0\uc11c \uace1\uc120 \\(y=x^{3\/2}\\)\uc758 \uae38\uc774\ub97c \uad6c\ud574 \ubcf4\uc790.<\/p>\n<p>\\(f(x) = x^{3\/2}\\)\uc774\ub77c\uace0 \ud558\uba74<br \/>\n\\[f &#8216; (x) = \\frac{3}{2}\\sqrt{x}\\]<br \/>\n\uadf8\ub9ac\uace0<br \/>\n\\[\\sqrt{1+(f'(x))^2} = \\sqrt{1+\\frac{9}{4}x}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uad6c\ud558\ub294 \uace1\uc120\uc758 \uae38\uc774\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\int_1^5 \\sqrt{1+\\frac{9}{4} x}\\,dx<br \/>\n&#038;= \\frac{4}{9} \\int_0^5 \\left( 1+ \\frac{9}{4}\\right)^{1\/2} \\cdot \\frac{9}{4} dx \\\\[5pt]<br \/>\n&#038;= \\frac{4}{9}\\left[ \\frac{2}{3} \\left( 1+ \\frac{9}{4} x\\right)^{3\/2} \\right]_0^5 = \\frac{335}{27}.<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 6.4.2.<\/span><\/p>\n<p>\\(0 \\le x \\le 1\\)\uc758 \ubc94\uc704\uc5d0\uc11c \uace1\uc120 \\(y=x^2\\)\uc758 \uae38\uc774\ub97c \uad6c\ud574 \ubcf4\uc790.<\/p>\n<p>\\(f(x)=x^2\\)\uc774\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74<br \/>\n\\[f &#8216; (x) = 2x\\]<br \/>\n\uadf8\ub9ac\uace0<br \/>\n\\[\\sqrt{1+(f'(x))^2} = \\sqrt{1+4x^2}\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uad6c\ud558\ub294 \uace1\uc120\uc758 \uae38\uc774\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\int_0^1\\sqrt{1+4x^2}\\,dx<br \/>\n&#038;= \\frac{1}{2}\\int_0^2 \\sqrt{1+u^2} \\,du \\\\[5pt]<br \/>\n&#038;= \\frac{1}{2}\\sqrt{5} + \\frac{1}{4} \\ln ( 2+\\sqrt{5} ).<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\ud68c\uc804\uba74\uc758 \ub113\uc774<\/h2>\n<p>\uace1\uc120\uc774 \uace1\uc120\uacfc \ub2ff\uc9c0 \uc54a\uc740 \uc9c1\uc120\uc744 \ucd95\uc73c\ub85c \ud68c\uc804\ud558\uc600\uc744 \ub54c \uc0dd\uae30\ub294 \uace1\uba74\uc744 <span class=\"defined\">\ud68c\uc804\uace1\uba74<\/span>(surface of revolution)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \ub2eb\ud78c\uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \uad6c\uac04 \\((a,\\,b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70, \\(a\\le x \\le b\\)\uc77c \ub54c \\(f(x) \\ge 0\\)\uc774\ub77c\uace0 \ud558\uc790. \\(a\\le x\\le b\\) \ubc94\uc704\uc5d0\uc11c \uace1\uc120 \\(y=f(x)\\)\ub97c \\(x\\)\ucd95\uc5d0 \ub300\ud558\uc5ec \ud55c \ubc14\ud034 \ud68c\uc804\uc2dc\ucf30\uc744 \ub54c \uc790\ucde8\uac00 \ub9cc\ub4dc\ub294 \uace1\uba74\uc758 \ub113\uc774 \\(S\\)\ub97c \uad6c\ud574 \ubcf4\uc790.<\/p>\n<p>\uad6c\uac04 \\([a,\\,b]\\)\uac00 \\(n\\)\uac1c\uc758 \uc18c\uad6c\uac04 \\([x_{k-1} ,\\, x_k ]\\)\ub4e4\ub85c \ub098\ub204\uc5b4\uc9c0\uace0, \uac01 \uc18c\uad6c\uac04\uc758 \uae38\uc774\ub97c \\(\\Delta x_k\\)\ub77c\uace0 \ud558\uc790. \uc18c\uad6c\uac04\uc758 \uc591\ucabd \ub05d\uc810\uc5d0 \ub300\uc751\ub418\ub294 \uace1\uc120 \uc704\uc758 \uc810 \\((x_{k-1} ,\\, f(x_{k-1}))\\)\uacfc \\((x_k ,\\, f(x_k))\\)\ub97c \uc774\uc740 \uc120\ubd84\uc744 \\(x\\)\ucd95\uc5d0 \ub300\ud558\uc5ec \ud55c \ubc14\ud034 \ud68c\uc804\uc2dc\ucf30\uc744 \ub54c \uc120\ubd84\uc758 \uc790\ucde8\uac00 \ub9cc\ub4dc\ub294 \uba74(\uc6d0\ubfd4\ub300\uc758 \uc606\uba74)\uc758 \ub113\uc774\ub97c \\(S_k\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(S_k\\)\uc758 \ud569\uc740 \\(S\\)\uc758 \uadfc\uc0bf\uac12\uc774\ub2e4.<\/p>\n<div style=\"margin-top: 1em; margin-bottom: 1.5em;\">\n<a href=\"\/blog\/wp-content\/uploads\/2021\/09\/ic_704_2.png\" data-rel=\"penci-gallery-image-content\" ><img decoding=\"async\" src=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_2.png\" alt=\"\" width=\"715\" height=\"398\" class=\"aligncenter size-full wp-image-7817\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_2.png 3575w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_2-300x167.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_2-1024x571.png 1024w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_2-768x428.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_2-1536x856.png 1536w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_2-2048x1141.png 2048w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_2-1920x1070.png 1920w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_2-1170x652.png 1170w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_704_2-585x326.png 585w\" sizes=\"(max-width: 715px) 100vw, 715px\" \/><\/a>\n<\/div>\n<p>\\(S_k\\)\ub294 \uc6d0\ubfd4\ub300\uc758 \uc606\uba74\uc758 \ub113\uc774\uc774\ubbc0\ub85c \ub2e4\uc74c\uacfc \uac19\uc774 \uacc4\uc0b0\ub41c\ub2e4.<br \/>\n\\[2\\pi \\left( \\frac{ f(x_{k-1}) + f(x_k) }{2} \\right) \\sqrt{ (\\Delta x_k )^2 + (\\Delta y_k )^2 } .\\]<br \/>\n\uc5ec\uae30\uc11c<br \/>\n\\[\\frac{ f(x_{k-1}) + f(x_k) }{2}\\]<br \/>\n\ub294 \\(f(x_{k-1})\\)\uc640 \\(f(x_k)\\) \uc0ac\uc774\uc758 \uac12\uc774\ubbc0\ub85c \uc5f0\uc18d\ud568\uc218\uc758 \uc0ac\uc787\uac12 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\frac{ f(x_{k-1}) + f(x_k) }{2} = f(x_k ^* )\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810 \\(x_k^*\\)\uac00 \uc5f4\ub9b0\uad6c\uac04 \\((x_{k-1},\\, x_k)\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \ub610\ud55c<br \/>\n\\[\\sqrt{ (\\Delta x_k )^2 + (\\Delta y_k )^2 } = \\sqrt{ 1+ \\left( \\frac{\\Delta y_k}{\\Delta x_k} \\right)^2 } \\Delta x_k\\]<br \/>\n\uc774\ub2e4. \ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\frac{\\Delta y_k}{\\Delta x_k} = f &#8216; (\\xi _k )\\]<br \/>\n\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810 \\(\\xi_k\\)\uac00 \uc5f4\ub9b0\uad6c\uac04 \\((x_{k-1} ,\\, x_k )\\)\uc5d0 \uc874\uc7ac\ud55c\ub2e4. \uc774\ub85c\uc368 \\(S_k\\)\ub4e4\uc758 \ud569\uc740<br \/>\n\\[\\sum_{k=1}^{n} 2\\pi f(x_k ^* )\\sqrt{1+ ( &#8216; (\\xi _k ))^2} \\Delta x_k\\]<br \/>\n\uc640 \uac19\uc774 \ud45c\ud604\ub41c\ub2e4. \uc810 \\(x_k\\)\ub4e4\ub85c \uc774\ub8e8\uc5b4\uc9c4 \ubd84\ud560\uc744 \\(P\\)\ub77c \ud558\uace0 \\(\\lVert P \\rVert \\rightarrow 0\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uba74 \\(x_k^*\\)\uc640 \\(\\xi_k\\)\uc758 \uac70\ub9ac\ub294 \\(0\\)\uc5d0 \uc218\ub834\ud558\uace0, \uc774\ub4e4\uc740 \ubaa8\ub450 \uc815\uc801\ubd84\uc758 \ubcc0\uc218 \\(x\\)\uac00 \ub41c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc704 \ud569\uc740 \ub2e4\uc74c \uc815\uc801\ubd84\uc5d0 \uc218\ub834\ud55c\ub2e4.<br \/>\n\\[2\\pi \\int_a^b f(x)\\sqrt{1+(f &#8216; (x))^2} \\,dx\\]<br \/>\n\uc9c0\uae08\uae4c\uc9c0 \ub2e4\uc74c \uacf5\uc2dd\uc744 \uc720\ub3c4\ud558\uc600\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 6.4.2. (\ud68c\uc804\uba74\uc758 \ub113\uc774)<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \ub2eb\ud78c\uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \uad6c\uac04 \\((a,\\,b)\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70, \\(a\\le x \\le b\\)\uc77c \ub54c \\(f(x) \\ge 0\\)\uc774\ub77c\uace0 \ud558\uc790. \\(a\\le x \\le b\\)\uc758 \ubc94\uc704\uc5d0\uc11c \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504\ub97c \\(x\\)\ucd95\uc5d0 \ub300\ud558\uc5ec \ud55c \ubc14\ud034 \ud68c\uc804\uc2dc\ucf30\uc744 \ub54c \uadf8\ub798\ud504\uc758 \uc790\ucde8\uac00 \ub9cc\ub4dc\ub294 \uace1\uba74\uc758 \ub113\uc774 \\(S\\)\ub294<br \/>\n\\[S = 2\\pi \\int_a^b f(x)\\sqrt{1+(f'(x))^2}\\,dx \\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 6.4.3.<\/span><br \/>\n\\(0\\le x\\le 3\\)\uc758 \ubc94\uc704\uc5d0\uc11c \ud3ec\ubb3c\uc120 \\(y^2 = 12x\\)\ub97c \\(x\\)\ucd95\uc5d0 \ub300\ud558\uc5ec \ud55c \ubc14\ud034 \ud68c\uc804\uc2dc\ucf30\uc744 \ub54c \uace1\uc120\uc758 \uc790\ucde8\uac00 \ub9cc\ub4dc\ub294 \uace1\uba74\uc758 \ub113\uc774 \\(S\\)\ub97c \uad6c\ud574 \ubcf4\uc790.<\/p>\n<p>\uc74c\ud568\uc218 \ubbf8\ubd84\ubc95\uc5d0 \uc758\ud558\uc5ec<br \/>\n\\[\\frac{dy}{dx} = \\frac{6}{y} ,\\quad 1+ \\left( \\frac{dy}{dx}\\right)^2 = \\frac{y^2 + 36}{y^2}\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uad6c\ud558\ub294 \uace1\uba74\uc758 \ub113\uc774\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\nS &#038;= 2\\pi \\int_0^3 y\\frac{\\sqrt{y^2 + 36}}{y} dx \\\\[5pt]<br \/>\n&#038;= 2\\pi \\int_0^3 \\sqrt{12x+36},dx \\\\[5pt]<br \/>\n&#038;= 24(2\\sqrt{2} -1) \\pi.<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<p>\ud3c9\uba74\uc5d0\uc11c \uace1\uc120\uc774 \ub9e4\uac1c\ubcc0\uc218 \ubc29\uc815\uc2dd\uc73c\ub85c \uc8fc\uc5b4\uc84c\uc744 \ub54c\uc5d0\ub3c4 \uace1\uc120\uc758 \uae38\uc774\ub97c \uad6c\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\uace1\uc120\uc774 \ub9e4\uac1c\ubcc0\uc218 \ubc29\uc815\uc2dd \\(x=f(t),\\) \\(y=g(t)\\)\ub85c \uc8fc\uc5b4\uc838 \uc788\ub2e4\uace0 \ud558\uc790. \\(t\\)\ub97c \uc2dc\uac01(time)\uc774\ub77c\uace0 \uc0dd\uac01\ud558\uba74 \uc810 \\((x,\\,y)\\)\uc5d0\uc11c \uc18d\ub825\uc740<br \/>\n\\[v(t) = \\sqrt{\\left(\\frac{dx}{dt}\\right)^2 + \\left(\\frac{dy}{dt}\\right)^2}\\]<br \/>\n\uc774\ub2e4. \ub530\ub77c\uc11c \\(t_1 \\le t \\le t_2\\)\uc77c \ub54c \\(x=f(t),\\) \\(y=g(t)\\)\ub85c \uc8fc\uc5b4\uc9c4 \uace1\uc120\uc758 \uae38\uc774\ub294<br \/>\n\\[\\int_{t_1}^{t_2} \\sqrt{ \\left(\\frac{dx}{dt}\\right)^2 + \\left(\\frac{dy}{dt}\\right)^2 } \\,dt\\]<br \/>\n\uc774\ub2e4. \uc774\uc640 \uac19\uc740 \uace1\uc120\uc744 \\(x\\)\ucd95\uc5d0 \ub300\ud558\uc5ec \ud55c \ubc14\ud034 \ud68c\uc804\uc2dc\ucf30\uc744 \ub54c \uc790\ucde8\uac00 \ub9cc\ub4dc\ub294 \uace1\uba74\uc758 \ub113\uc774 \\(S\\)\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[S = 2\\pi \\int_{t_1}^{t_2} y\\sqrt{ \\left( \\frac{dx}{dt}\\right)^2 + \\left(\\frac{dy}{dt}\\right)^2 } \\,dt.\\]<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 6.4.4.<\/span><\/p>\n<p>\\(a\\)\uac00 \uc591\uc218\uc778 \uc0c1\uc218\ub77c\uace0 \ud558\uc790.<br \/>\n\\[x = a\\cos^3 \\theta,\\quad y=a\\sin^3\\theta\\]<br \/>\n\ub85c \uc8fc\uc5b4\uc9c4 \ud558\uc774\ud3ec\uc0ac\uc774\ud074\ub85c\uc774\ub4dc(hypocycloid)\ub97c \\(x\\)\ucd95\uc5d0 \ub300\ud558\uc5ec \ud55c \ubc14\ud034 \ud68c\uc804\uc2dc\ucf30\uc744 \ub54c \uace1\uc120\uc758 \uc790\ucde8\uac00 \ub9cc\ub4dc\ub294 \uace1\uba74\uc758 \ub113\uc774 \\(S\\)\ub97c \uad6c\ud574 \ubcf4\uc790.<\/p>\n<p>\\(0 \\le \\theta \\le \\pi\\)\uc778 \ubc94\uc704\uc5d0\uc11c \uad6c\ud558\uba74 \ucda9\ubd84\ud558\ub2e4. (\uadf8\ub798\ud504\ub97c \uadf8\ub824 \ubcf4\uc790.)<br \/>\n\\[\\frac{dx}{d\\theta} = -3a\\cos^2\\theta \\,\\sin\\theta ,\\quad \\frac{dy}{d\\theta} = 3a\\sin^2 \\theta\\,\\cos\\theta\\]<br \/>\n\uc774\ubbc0\ub85c<br \/>\n\\[\\left( \\frac{dx}{d\\theta} \\right)^2 + \\left(\\frac{dy}{d\\theta} \\right)^2 = 9a^2 \\cos^2 \\theta\\,\\sin^2 \\theta\\]<br \/>\n\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uad6c\ud558\ub294 \uace1\uba74\uc758 \ub113\uc774\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[S = 2 \\cdot 2\\pi \\int_0^{\\pi\/2} (a\\sin^3 \\theta ) \\cdot 3a \\,\\cos\\theta\\,\\sin\\theta\\,d\\theta = \\frac{12a^2 \\pi}{5}.\\]\n<\/p>\n<\/div>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc785\uccb4\ub3c4\ud615\uc758 \ubd80\ud53c<\/h2>\n<p>\ud3c9\uba74\uc5d0\uc11c \ud568\uc218\uc758 \uadf8\ub798\ud504\ub85c \ub458\ub7ec\uc2f8\uc778 \ubd80\ubd84\uc758 \ub113\uc774\ub97c \uad6c\ud560 \ub54c \uc801\ubd84\uc744 \uc0ac\uc6a9\ud558\ub294 \uac83\ucc98\ub7fc 3\ucc28\uc6d0 \uacf5\uac04\uc5d0\uc11c \uadf8\ub798\ud504\ub85c \ub458\ub7ec\uc2f8\uc778 \ubd80\ubd84\uc758 \ubd80\ud53c\ub97c \uad6c\ud560 \ub54c\ub3c4 \uc801\ubd84\uc744 \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 6.4.3. (\uc785\uccb4\ub3c4\ud615\uc758 \ubd80\ud53c)<\/span><\/p>\n<p>\\(\\mathbb{R}^3\\) \uacf5\uac04\uc5d0\uc11c \\(a\\le x\\le b\\)\uc758 \ubc94\uc704\uc5d0 \ub193\uc5ec \uc788\uace0 \uc720\uacc4\uc778 \uc785\uccb4\ub3c4\ud615\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uc774 \ub3c4\ud615\uc744 \ud3c9\uba74 \\(x=t\\)\ub85c \uc798\ub790\uc744 \ub54c \ub2e8\uba74\uc758 \ub113\uc774\ub97c \\(S(t)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \uc774 \uc785\uccb4\ub3c4\ud615\uc758 \ubd80\ud53c \\(V\\)\ub294<br \/>\n\\[V = \\int_a^b S(x) dx\\]<br \/>\n\uc774\ub2e4. \ub9cc\uc57d \uc774 \uc785\uccb4\ub3c4\ud615\uc774 \\(a \\le x \\le b\\)\uc758 \ubc94\uc704\uc5d0\uc11c \ud568\uc218 \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504\ub97c \\(x\\)\ucd95\uc73c\ub85c \ud68c\uc804\uc2dc\ucf30\uc744 \ub54c \uace1\uc120\uc758 \uc790\ucde8\uac00 \ub9cc\ub4dc\ub294 \uba74\uc73c\ub85c \ub458\ub7ec\uc2f8\uc778 \ubaa8\uc591\uc774\ub77c\uba74 \uc774 \uc785\uccb4\ub3c4\ud615\uc758 \ubd80\ud53c \\(V\\)\ub294<br \/>\n\\[V = \\pi \\int_a^b (f(x))^2 \\,dx\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 6.4.5.<\/span><\/p>\n<p>\ud3ec\ubb3c\uc120 \\(y=4x^2\\)\uacfc \ub450 \uc9c1\uc120 \\(x=0,\\) \\(y=16\\)\uc73c\ub85c \ub458\ub7ec\uc2f8\uc778 \uc601\uc5ed\uc744 \\(x\\)\ucd95\uc5d0 \ub300\ud558\uc5ec \ud55c \ubc14\ud034 \ud68c\uc804\uc2dc\ucf30\uc744 \ub54c \uc790\ucde8\uac00 \ub9cc\ub4dc\ub294 \uc785\uccb4\ub3c4\ud615\uc758 \ubd80\ud53c\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\pi\\int_0^{16} x^2 \\,dy = \\pi \\int_0^{16} \\frac{y}{4}\\,dy = \\frac{\\pi}{8}\\times 256 = 32\\pi.\\]\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 6.4.6.<\/span><\/p>\n<p>\ubc11\uba74\uc758 \ubc18\uc9c0\ub984\uc774 \\(r\\)\uc774\uace0 \ub192\uc774\uac00 \\(h\\)\uc778 \uc6d0\ubfd4\uc758 \ubd80\ud53c\ub97c \uad6c\ud574 \ubcf4\uc790. \uc774 \uc6d0\ubfd4\uc740 \uc9c1\uc120 \\(y=(r\/h)x\\)\uc640 \\(x\\)\ucd95, \uadf8\ub9ac\uace0 \ub450 \uc9c1\uc120 \\(x=0,\\) \\(x=h\\)\ub85c \ub458\ub7ec\uc2f8\uc778 \ub3c4\ud615\uc744 \\(x\\)\ucd95\uc5d0 \ub300\ud558\uc5ec \ud55c \ubc14\ud034 \ud68c\uc804\uc2dc\ucf30\uc744 \ub54c \uc790\ucde8\uc640 \uac19\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc774 \uc6d0\ubfd4\uc758 \ubd80\ud53c\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<br \/>\n\\[\\begin{align}<br \/>\n\\pi\\int_0^h y^2\\,dx<br \/>\n&#038;= \\pi \\int_0^h  \\frac{r^2}{h^2} x^2 \\,dx \\\\[5pt]<br \/>\n&#038;= \\left[ \\frac{\\pi r^2}{h^2} \\cdot \\frac{1}{3} x^3 \\right]_{x=0}^{x=h} \\\\[5pt]<br \/>\n&#038;= \\frac{\\pi r^2}{h^2} \\cdot \\frac{1}{3} h^3 \\\\[5pt]<br \/>\n&#038;= \\frac{1}{3} \\pi r^2 h.<br \/>\n\\end{align}\\]\n<\/p>\n<\/div>\n<p><!-- ##################################################################### --><br \/>\n<!--\n\n\n<h2 class=\"itc_h2\">\uc81c\ubaa9<\/h2>\n\n\n\n\n\n<p>.<\/p>\n\n\n\n\n\n<p>.<\/p>\n\n\n--><\/p>\n<p><!--\n\n\n\n<div class=\"theorem margintop2\">\n\n\n<p><span class=\"theorem\">\uc815\ub9ac 1.<\/span>\n\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<div class=\"proof\">\n\n\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n\n\n\n\n<p>\n\n<span class=\"qed\"><\/span><\/p>\n\n<\/div>\n\n\n\n\n########\n\n\u201c\u201d\n\u2018\u2019\n\n\n\n<div class=\"example\">\n\n\n<p><span class=\"example\">\ubcf4\uae30 1.<\/span>\n\n<\/p>\n\n<\/div>\n\n\n\n\n\n<h2 class=\"itc_h2\"><\/h2>\n\n\n\n--><\/p>\n<p><!--\n\n\n<div style=\"display: none; visibility: hidden;\">\n\\[\n\\newcommand{\\Hom}{{\\operatorname{Hom}}}\n\\newcommand{\\Mat}{{\\operatorname{Mat}}}\n\\newcommand{\\proj}{{\\operatorname{proj}}}\n\\newcommand{\\adj}{{\\operatorname{adj}}}\n\\]\n<\/div>\n\n\n--><\/p>\n<div class=\"box itc_prev_next_box\">\n<ul class=\"itc_ul\">\n<li class=\"itc_li_prev\">\uc55e\uc758 \uae00 : <a href=\"..\/integral-formulas\">\uc801\ubd84 \uacf5\uc2dd<\/a><\/li>\n<p><!-- li class=\"itc_li_next\">\ub2e4\uc74c \uae00 : <a href=\"..\/improper-integrals\">\uc774\uc0c1\uc801\ubd84<\/a><\/li -->\n<\/ul>\n<\/div>\n","protected":false},"excerpt":{"rendered":"<p>\uc774 \uae00\uc740 \u300e\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c\u300f 6\uc7a5 4\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30) \uc774 \uc808\uc5d0\uc11c\ub294 \uc801\ubd84\uc744 \uc774\uc6a9\ud558\uc5ec \uace1\uc120\uc758 \uae38\uc774, \ud68c\uc804\uba74\uc758 \ub113\uc774, \uc785\uccb4\ub3c4\ud615\uc758 \ubd80\ud53c \uad6c\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf4\uc790. \uace1\uc120\uc758 \uae38\uc774 \ud568\uc218 \\(f\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub41c \ubbf8\ubd84 \uac00\ub2a5\ud55c \uc2e4\uc22b\uac12 \ud568\uc218\ub77c\uace0 \ud558\uc790. \uc774\uc81c \\(a\\le x le b\\)\uc758 \ubc94\uc704\uc5d0\uc11c \\(y=f(x)\\)\uc758 \uadf8\ub798\ud504\uc758 \uae38\uc774\ub97c \uad6c\ud574 \ubcf4\uc790. \uad6c\uac04 \\([a,\\,b]\\)\ub97c \\(n\\)\uac1c\uc758 \uc18c\uad6c\uac04 \\(\\left[ x_{i-1},\\, x_i \\right]\\)\ub85c \ubd84\ud560\ud558\uc790. \uc989 \\(a = x_0 < x_1&hellip;\n<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":6620,"menu_order":704,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"class_list":["post-6739","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6739","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/types\/page"}],"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=6739"}],"version-history":[{"count":29,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6739\/revisions"}],"predecessor-version":[{"id":8379,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6739\/revisions\/8379"}],"up":[{"embeddable":true,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6620"}],"wp:attachment":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/media?parent=6739"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}