{"id":6733,"date":"2021-07-21T00:09:06","date_gmt":"2021-07-20T15:09:06","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=6733"},"modified":"2022-03-06T19:47:33","modified_gmt":"2022-03-06T10:47:33","slug":"definite-integrals","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-calculus\/definite-integrals\/","title":{"rendered":"\uc815\uc801\ubd84"},"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 1\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><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc815\uc801\ubd84\uc758 \uc815\uc758<\/h2>\n<p>\\(a < b\\)\uc774\uace0 \\(f\\)\uac00 \\(I=[a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc720\uacc4\uc778 \uc2e4\uc22b\uac12 \ud568\uc218\ub77c\uace0 \ud558\uc790.<\/p>\n<p>\\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba70 \uc784\uc758\uc758 \\(x\\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \\ge 0\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \uace1\uc120 \\(y=f(x)\\)\uc640 \\(x\\)\ucd95, \ub450 \uc9c1\uc120 \\(x=a,\\) \\(x=b\\)\ub85c \ub458\ub7ec\uc2f8\uc778 \ubd80\ubd84\uc758 \ub113\uc774\ub97c \\(A\\)\ub77c\uace0 \ud558\uc790. \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ub2e4\uc74c\uacfc \uac19\uc740 \\((n+1)\\)\uac1c\uc758 \uc810 \\(x_i\\)\ub97c \ud0dd\ud558\uc790.<br \/>\n\\[a = x_0 < x_1 < x_2 < \\cdots < x_{i-1} < x_i < \\cdots < x_{n-1} < x_n = b.\\]\n\uc774\ub7ec\ud55c \uc810\ub4e4\uc758 \ubaa8\uc784\uc744 \\(P\\)\ub77c\uace0 \ud558\uc790. \uc989\n\\[P = \\left\\{ x_0 ,\\, x_1 ,\\, x_2 ,\\, \\cdots ,\\, x_n \\right\\}\\]\n\uc774\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(P\\)\ub97c \\(I\\)\uc758 <span class=\"defined\">\ubd84\ud560<\/span>(partition)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc774\uc640 \uac19\uc774 \uc8fc\uc5b4\uc9c4 \ubd84\ud560 \\(P\\)\uc758 \uc810\uc744 \uc774\uc6a9\ud558\uc5ec \\(I\\)\ub97c \\(n\\)\uac1c\uc758 \uc18c\uad6c\uac04\uc73c\ub85c \uc790\ub97c \uc218 \uc788\ub2e4:<br \/>\n\\[ \\left[ x_0 ,\\,x_1 \\right] ,\\,\\, \\left[ x_1 ,\\,x_2 \\right] ,\\,\\, \\cdots ,\\,\\, \\left[ x_{i-1} ,\\,x_i \\right] ,\\,\\, \\cdots ,\\,\\, \\left[ x_{n-1} ,\\,x_n \\right] .\\]<br \/>\n\\(i\\)\ubc88\uc9f8 \uc18c\uad6c\uac04\uc758 \uae38\uc774\ub97c \\(\\Delta x_i\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \uc989 \\(\\Delta x_i = x_i &#8211; x_{i-1}\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n<div class=\"margintop1 marginbottom1\">\n<a href=\"\/blog\/wp-content\/uploads\/2021\/09\/ic_701_1.png\" data-rel=\"penci-gallery-image-content\" ><img fetchpriority=\"high\" decoding=\"async\" src=\"\/blog\/wp-content\/uploads\/2021\/09\/ic_701_1.png\" alt=\"\" width=\"559\" height=\"182\" class=\"aligncenter size-full wp-image-7812\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_701_1.png 1676w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_701_1-300x98.png 300w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_701_1-1024x334.png 1024w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_701_1-768x250.png 768w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_701_1-1536x500.png 1536w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_701_1-1170x381.png 1170w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_701_1-585x191.png 585w\" sizes=\"(max-width: 559px) 100vw, 559px\" \/><\/a>\n<\/div>\n<p>\uc704 \uadf8\ub9bc\uacfc \uac19\uc774 \uc18c\uad6c\uac04 \\(\\left[ x_{i-1} ,\\, x_i \\right]\\)\uc5d0\uc11c \\(f\\)\uc758 \ucd5c\uc19f\uac12\uacfc \ucd5c\ub313\uac12\uc744 \uac01\uac01 \\(m_i ,\\) \\(M_i\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[m_1 \\Delta x_1 + m_2 \\Delta x_2 + \\cdots + m_n \\Delta x_n<br \/>\n\\le A \\le<br \/>\nM_1 \\Delta x_1 + M_2 \\Delta x_2 + \\cdots + M_n \\Delta x_n . \\tag{*}\\]<br \/>\n\uc704 \ubd80\ub4f1\uc2dd\uc5d0\uc11c \uac00\uc7a5 \uc67c\ucabd\uc758 \ud569\uc744 \\(L(f,\\,P)\\)\ub85c, \uac00\uc7a5 \uc624\ub978\ucabd\uc758 \ud569\uc744 \\(U(f,\\,P)\\)\ub85c \ub098\ud0c0\ub0b4\uc790. \uadf8\ub7ec\uba74 \uc704 \ubd80\ub4f1\uc2dd\uc740 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0bc \uc218 \uc788\ub2e4.<br \/>\n\\[L(f,\\,P) \\le A \\le U(f,\\,P).\\]<br \/>\n\ubd84\ud560 \\(P\\)\uac00 \uc8fc\uc5b4\uc84c\uc744 \ub54c \\(\\Delta x_i\\) \uc911 \uac00\uc7a5 \ud070 \uac12\uc744 \\(P\\)\uc758 <span class=\"defined\">\ub178\ub984<\/span>(norm)\uc774\ub77c\uace0 \ubd80\ub974\uace0 \\(\\lVert P \\rVert\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \ub9cc\uc57d \\(\\lVert P \\rVert \\rightarrow 0\\)\uc77c \ub54c \\(L(f,\\,P)\\)\uac00 \ud558\ub098\uc758 \uac12 \\(I_L\\)\uc5d0 \ud55c\uc5c6\uc774 \uac00\uae4c\uc6cc\uc9c0\uba74 \\(I_L\\)\uc744 \\(I\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\ud558\uc801\ubd84<\/span>(lower integral)\uc774\ub77c\uace0 \ubd80\ub974\uace0<br \/>\n\\[\\underline \\int_{a}^{b} f(x) dx\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4. \ub9c8\ucc2c\uac00\uc9c0\ub85c \ub9cc\uc57d \\(\\lVert P \\rVert \\rightarrow 0\\)\uc77c \ub54c \\(U(f,\\,P)\\)\uac00 \ud558\ub098\uc758 \uac12 \\(I_U\\)\uc5d0 \ud55c\uc5c6\uc774 \uac00\uae4c\uc6cc\uc9c0\uba74 \\(I_U\\)\ub97c \\(I\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\uc0c1\uc801\ubd84<\/span>(upper integral)\uc774\ub77c\uace0 \ubd80\ub974\uace0<br \/>\n\\[\\overline \\int_{a}^{b} f(x) dx\\]<br \/>\n\uc640 \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<\/p>\n<p>\ub9cc\uc57d \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \ud558\uc801\ubd84\uacfc \uc0c1\uc801\ubd84\uc774 \uc77c\uce58\ud558\uba74 \u201c\\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c <span class=\"defined\">\uc801\ubd84 \uac00\ub2a5\ud558\ub2e4<\/span>\u201d\ub77c\uace0 \ub9d0\ud558\uace0, \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \ud558\uc801\ubd84\uacfc \uc0c1\uc801\ubd84\uc744 \u2018\\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\ub9ac\ub9cc \uc801\ubd84<\/span>(Riemann integral)\u2019 \ub610\ub294 \uac04\ub2e8\ud788 \u2018\\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 <span class=\"defined\">\uc801\ubd84<\/span>\u2019\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc801\ubd84\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.<br \/>\n\\[\\int_a^b f(x) dx .\\]<br \/>\n\uc774\uc640 \uac19\uc740 \ud45c\ud604\uc5d0\uc11c \\(x\\)\ub294 \ub3c5\ub9bd\ubcc0\uc218\uac00 \uc544\ub2c8\ub2e4. \uc608\ucee8\ub300 \ub2e4\uc74c \uc2dd\uc740 \ubaa8\ub450 \uac19\uc740 \uc801\ubd84\uc744 \ub098\ud0c0\ub0b8\ub2e4.<br \/>\n\\[\\int_a^b f(t) dt ,\\quad \\int_a^b f(s) ds ,\\quad \\int_a^b f(z)dz .\\]<br \/>\n\uc801\ubd84\uc758 \uac12\uc740 \\(a,\\) \\(b,\\) \\(f\\)\uc5d0 \uc758\ud558\uc5ec \uacb0\uc815\ub41c\ub2e4. \ub9cc\uc57d \\(a=b\\)\uc774\uba74<br \/>\n\\[\\int_a^a f(x)dx =0\\]<br \/>\n\uc774\ub77c\uace0 \uc815\uc758\ud55c\ub2e4. \ub9cc\uc57d \\(a > b\\)\uc774\uba74<br \/>\n\\[\\int_a^b f(x)dx =-\\int _b^a f(x)dx\\]<br \/>\n\ub77c\uace0 \uc815\uc758\ud55c\ub2e4.<\/p>\n<p>\uc9c0\uae08\uae4c\uc9c0 \uc801\ubd84\uc744 \uc815\uc758\ud55c \uacfc\uc815\uc744 \uc0b4\uc9dd \uc218\uc815\ud558\uba74 \\(f\\)\uac00 \uc5f0\uc18d\uc774 \uc544\ub2cc \ud568\uc218\uc77c \ub54c\ub3c4 \uc801\ubd84\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \ub9cc\uc57d \\(f\\)\uac00 \uc18c\uad6c\uac04 \\(\\left[ x_{i-1} ,\\,x_i \\right]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774 \uc544\ub2c8\ub77c\uba74 \uc774 \uad6c\uac04\uc5d0\uc11c \\(f\\)\ub294 \ucd5c\ub313\uac12\uc774\ub098 \ucd5c\uc19f\uac12\uc744 \uac16\uc9c0 \uc54a\uc744 \uc218\ub3c4 \uc788\ub2e4. \ud558\uc9c0\ub9cc \ub9cc\uc57d \\(f\\)\uac00 \uc774 \uad6c\uac04\uc5d0\uc11c \uc720\uacc4\ub77c\uba74<\/p>\n<p class=\"aligncenter\">\u201c\uc784\uc758\uc758 \\(x\\in \\left[ x_{i-1} ,\\, x_i \\right]\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \\le M_i\\)\u201d<\/p>\n<p>\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(M_i\\) \uc911\uc5d0\uc11c \uac00\uc7a5 \uc791\uc740 \uac12\uc744 \ucde8\ud558\uace0,<\/p>\n<p class=\"aligncenter\">\u201c\uc784\uc758\uc758 \\(x\\in \\left[ x_{i-1} ,\\, x_i \\right]\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \\ge m_i\\)\u201d<\/p>\n<p>\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \\(m_i\\) \uc911\uc5d0\uc11c \uac00\uc7a5 \ud070 \uac12\uc744 \ucde8\ud560 \uc218 \uc788\ub2e4.<\/p>\n<p>\uc801\ubd84\uc744 \uc815\uc758\ud55c \uacfc\uc815\uc744 \uc870\uc2ec\uc2a4\ub7fd\uac8c \uc0b4\ud3b4\ubcf4\uba74 \\(f(x) \\ge 0\\)\uc774\ub77c\ub294 \uc870\uac74\uc740 \uc0ac\uc6a9\ub418\uc9c0 \uc54a\uc558\ub2e4. \uc774 \uc870\uac74\uc740 \ub2e8\uc9c0 \u2018\ub113\uc774\u2019\ub77c\ub294 \uc9c1\uad00\uc801 \uac1c\ub150\uc744 \ubc14\ud0d5\uc73c\ub85c \uc2dc\uac01\uc801 \uc2ec\uc0c1\uc744 \ud615\uc131\ud558\uae30 \uc704\ud574 \ud544\uc694\ud588\uc744 \ubfd0\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(f(x) \\ge 0\\)\uc774\ub77c\ub294 \uc870\uac74 \ub610\ud55c \uc801\ubd84\uc744 \uc815\uc758\ud558\ub294 \ub370\uc5d0 \ud544\uc694\ud558\uc9c0 \uc54a\ub2e4.<\/p>\n<p>\uc774\ub85c\uc368 \\(f\\)\uac00 \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uc77c \ub54c \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc801\ubd84\uc774 \uc815\uc758\ub418\uc5c8\ub2e4.<\/p>\n<p><!-- ##################################################################### --><\/p>\n<h2 class=\"itc_h2\">\uc815\uc801\ubd84\uc758 \uae30\ubcf8 \uc131\uc9c8<\/h2>\n<p>\uc774\uc81c \ub2e4\uc74c\uacfc \uac19\uc740 \ub450 \uac00\uc9c0 \uc758\ubb38\uc774 \uc0dd\uae34\ub2e4.<\/p>\n<ul>\n<li>\uc5b4\ub5a0\ud55c \ud568\uc218\uac00 \uc801\ubd84 \uac00\ub2a5\ud55c\ub2e4?<\/li>\n<li>\ub9cc\uc57d \uc801\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uac00 \uc8fc\uc5b4\uc9c4\ub2e4\uba74 \uc801\ubd84\uc744 \uc5b4\ub5bb\uac8c \uacc4\uc0b0\ud558\ub294\uac00?<\/li>\n<\/ul>\n<p>\ub2e4\uc74c \uc815\ub9ac\ub294 \uccab \ubc88\uc9f8 \uc9c8\ubb38\uc5d0 \ub300\ud55c \ubd80\ubd84\uc801\uc778 \ub2f5\uc744 \uc81c\uc2dc\ud55c\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 6.1.1. (\ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131)<\/span><\/p>\n<p>\\(I\\)\uac00 \uae38\uc774\uac00 \uc591\uc218\uc778 \ub2eb\ud78c\uad6c\uac04\uc774\uace0 \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc720\uacc4\uc778 \uc2e4\uc22b\uac12 \ud568\uc218\ub77c\uace0 \ud558\uc790.<\/p>\n<ol class=\"bracket\">\n<li>\ub9cc\uc57d \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/li>\n<li>\ub9cc\uc57d \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \uac1c\uc218\uac00 \uc720\ud55c\uc774\uba74 \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/li>\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\"><span class=\"proof\">\uc99d\uba85<\/span><\/p>\n<p>\ubbf8\uc801\ubd84\ud559\uc744 \ucc98\uc74c \uacf5\ubd80\ud55c\ub2e4\uba74 \uc774 \uc815\ub9ac\uc758 \uc99d\uba85\uc744 \uc0dd\ub7b5\ud574\ub3c4 \uc88b\ub2e4. \uadf8\ub798\ub3c4 \ubcf4\uace0 \uc2f6\ub2e4\uba74 [1]\uc758 \uc99d\uba85\uc740 <a href=\"\/blog\/articles\/calculus-the-definite-integral\/\">\uc815\uc801\ubd84\uc758 \uc815\uc758<\/a> \uc815\ub9ac 5\ub97c \ubcf4\uae30 \ubc14\ub780\ub2e4. [2]\uc758 \uc99d\uba85\uc740 <a href=\"\/blog\/articles\/calculus-the-definite-integral\/\">\uc815\uc801\ubd84\uc758 \uc815\uc758<\/a> \uc815\ub9ac 7\uc744 \ubcf4\uae30 \ubc14\ub780\ub2e4.\n<\/p>\n<\/div>\n<p>\uc801\ubd84 \uac00\ub2a5\uc131\uacfc \uad00\ub828\ub41c \uc704\uc640 \uac19\uc740 \uc870\uac74\uc774 \\(f\\)\uac00 \uc801\ubd84 \uac00\ub2a5\ud558\uae30 \uc704\ud55c \ucda9\ubd84\uc870\uac74\uc744 \uc81c\uacf5\ud558\uae30\ub294 \ud558\uc9c0\ub9cc \ud544\uc694\ucda9\ubd84\uc870\uac74\uc744 \uc81c\uacf5\ud558\uc9c0\ub294 \uc54a\ub294\ub2e4. \ud558\uc9c0\ub9cc \ubbf8\uc801\ubd84\ud559 \uc785\ubb38 \uc218\uc900\uc5d0\uc11c\ub294 \uc704 \uc815\ub9ac\ub9cc\uc73c\ub85c \uc801\ubd84\uc758 \uc131\uc9c8\uc744 \ub17c\ud558\ub294 \ub370\uc5d0 \ucda9\ubd84\ud558\ub2e4.<\/p>\n<p>\uc0ac\uc2e4 \ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\uc758 \uce21\ub3c4\uac00 \\(0\\)\uc778 \uac83\uc774\ub2e4. \uc774\uc640 \uac19\uc740 \uc815\ub9ac\ub97c <span class=\"defined\">\ub974\ubca0\uadf8\uc758 \uc815\ub9ac<\/span>(Lebesgue&#8217;s theorem)\ub77c\uace0 \ubd80\ub978\ub2e4. \ub974\ubca0\uadf8\uc758 \uc815\ub9ac\ub97c \uae4a\uc774 \uacf5\ubd80\ud558\uace0 \uc2f6\uc740 \uc0ac\ub78c\uc740 \uc774 \ube14\ub85c\uadf8\uc758 \uae00 <a href=\"\/blog\/articles\/calculus-lebesgue-theorem-for-riemann-integrability\/\">\uc801\ubd84 \uac00\ub2a5\uc131\uc5d0 \ub300\ud55c \ub974\ubca0\uadf8\uc758 \uc815\ub9ac<\/a>\ub97c \ubcf4\uae30 \ubc14\ub780\ub2e4.<\/p>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 6.1.1.<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\(I=[0,\\,1]\\)\uc5d0\uc11c \\(f(x)=x\\)\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<p>\\(n\\)\uc774 (\ucda9\ubd84\ud788 \ud070) \uc790\uc5f0\uc218\ub77c\uace0 \ud558\uace0, \ub2e4\uc74c\uacfc \uac19\uc740 \uc810\uc744 \uc0dd\uac01\ud558\uc790.<br \/>\n\\[x_i = \\frac{i}{n} ,\\quad (i=0,\\,1,\\,2,\\,\\cdots ,\\,n ).\\]<br \/>\n\uc18c\uad6c\uac04 \\(\\left[ x_{i-1} ,\\, x_i \\right]\\)\uc5d0\uc11c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\Delta x_i = \\frac{1}{n} ,\\quad m_i = f\\left( x_{i-1} \\right) = \\frac{i-1}{n} ,\\quad M_i = f\\left( x_i \\right) = \\frac{i}{n}.\\]<br \/>\n\\(I\\)\uc5d0\uc11c \\(f\\)\uc758 \uc801\ubd84\uc758 \uac12\uc744 \\(A\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\left\\{ \\frac{0}{n} + \\frac{1}{n} + \\cdots + \\frac{n-1}{n} \\right\\} \\frac{1}{n}<br \/>\n\\le A \\le<br \/>\n\\left\\{ \\frac{1}{n} + \\frac{2}{n} + \\frac{3}{n} + \\cdots + \\frac{n}{n} \\right\\} \\frac{1}{n}.\\]<br \/>\n\uc774 \uc0c1\ud669\uc744 \uadf8\ub9bc\uc73c\ub85c \ub098\ud0c0\ub0b4\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n<div class=\"margintop1 marginbottom1\">\n<a href=\"\/blog\/wp-content\/uploads\/2021\/09\/ic_701_3.png\" data-rel=\"penci-gallery-image-content\" ><img decoding=\"async\" src=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_701_3.png\" alt=\"\" width=\"170\" height=\"173\" class=\"aligncenter size-full wp-image-7813\" srcset=\"https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_701_3.png 511w, https:\/\/sasamath.com\/blog\/wp-content\/uploads\/2021\/09\/ic_701_3-296x300.png 296w\" sizes=\"(max-width: 170px) 100vw, 170px\" \/><\/a>\n<\/div>\n<p>\ubd80\ub4f1\uc2dd\uc5d0\uc11c \ud569\uc744 \uacc4\uc0b0\ud558\uba74<br \/>\n\\[\\frac{n(n-1)}{2n^2} \\le A \\le \\frac{n(n+1)}{2n^2}\\]<br \/>\n\uc774\uba70, \uc2dd\uc744 \ubcc0\ud615\ud558\uba74<br \/>\n\\[\\frac{1}{2} &#8211; \\frac{1}{2n} \\le A \\le  \\frac{1}{2} + \\frac{1}{2n}\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4. \\(n\\rightarrow\\infty\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uba74 \\(\\Delta x_i \\rightarrow 0\\)\uc774\uba70,<br \/>\n\\[\\frac{1}{2} \\le A \\le \\frac{1}{2}\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\int_0^1 f(x) dx = \\int_0^1 x \\,dx = \\frac{1}{2}\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 6.1.2.<\/span><\/p>\n<p>\ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\(I=[0,\\,1]\\)\uc5d0\uc11c \\(f(x)=x^2\\)\uc73c\ub85c \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \\(f\\)\ub294 \\(I\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<p>\\(n\\)\uc774 (\ucda9\ubd84\ud788 \ud070) \uc790\uc5f0\uc218\ub77c\uace0 \ud558\uace0, \ub2e4\uc74c\uacfc \uac19\uc740 \uc810\uc744 \uc0dd\uac01\ud558\uc790.<br \/>\n\\[x_i = \\frac{i}{n} ,\\quad (i=0,\\,1,\\,2,\\,\\cdots ,\\,n ).\\]<br \/>\n\uc18c\uad6c\uac04 \\(\\left[ x_{i-1} ,\\, x_i \\right]\\)\uc5d0\uc11c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\Delta x_i = \\frac{1}{n} ,\\quad m_i = f\\left( x_{i-1} \\right) = \\frac{(i-1)^2}{n^2} ,\\quad M_i = f\\left( x_i \\right) = \\frac{i^2}{n^2}.\\]<br \/>\n\\(I\\)\uc5d0\uc11c \\(f\\)\uc758 \uc801\ubd84\uc758 \uac12\uc744 \\(A\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \ub2e4\uc74c \ubd80\ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.<br \/>\n\\[\\left\\{ \\frac{0^2}{n^2} + \\frac{1^2}{n^2} + \\cdots + \\frac{(n-1)^2}{n^2} \\right\\} \\frac{1}{n}<br \/>\n\\le A \\le<br \/>\n\\left\\{ \\frac{1^2}{n^2} + \\frac{2^2}{n^2} + \\frac{3^2}{n^2} + \\cdots + \\frac{n^2}{n^2} \\right\\} \\frac{1}{n}.\\]<br \/>\n\ubd80\ub4f1\uc2dd\uc5d0\uc11c \ud569\uc744 \uacc4\uc0b0\ud558\uba74<br \/>\n\\[\\frac{n(n-1)(2n-1)}{6n^3} \\le A \\le \\frac{n(n+1)(2n+1)}{6n^3}\\]<br \/>\n\uc774\ub2e4. \\(n\\rightarrow\\infty\\)\uc778 \uadf9\ud55c\uc744 \ucde8\ud558\uba74 \\(\\Delta x_i \\rightarrow 0\\)\uc774\uba70,<br \/>\n\\[\\frac{2}{6} \\le A \\le \\frac{2}{6}\\]<br \/>\n\uc744 \uc5bb\ub294\ub2e4. \uadf8\ub7ec\ubbc0\ub85c<br \/>\n\\[\\int_0^1 x^2 \\,dx = \\frac{1}{3}\\]<br \/>\n\uc774\ub2e4.\n<\/p>\n<\/div>\n<p>\uc815\uc801\ubd84\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uc131\uc9c8\uc744 \uac00\uc9c4\ub2e4.<\/p>\n<div class=\"theorem margintop2\">\n<p><span class=\"theorem\">\uc815\ub9ac 6.1.2. (\uc815\uc801\ubd84\uc758 \uc131\uc9c8)<\/span><\/p>\n<p>\\(f\\)\uc640 \\(g\\)\uac00 \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc720\uacc4\uc778 \uc2e4\uc22b\uac12 \ud568\uc218\ub77c\uace0 \ud558\uc790. \\(f\\)\uc640 \\(g\\)\uac00 \ubaa8\ub450 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \uac00\uc815\ud558\uace0, \\(k\\)\uac00 \uc2e4\uc218\ub77c\uace0 \ud558\uc790. \uadf8\ub7ec\uba74 \\(f+g,\\) \\(f-g,\\) \\(kf\\)\uac00 \ubaa8\ub450 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uba70, \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<ol class=\"bracket\">\n<li>\\(\\displaystyle \\int_a^b \\left\\{ f(x) + g(x) \\right\\} dx = \\int_a^b f(x) dx + \\int_a^b g(x) dx.  \\)<\/li>\n<li class=\"margintop1\">\\(\\displaystyle \\int_a^b \\left\\{ f(x) &#8211; g(x) \\right\\} dx = \\int_a^b f(x) dx &#8211; \\int_a^b g(x) dx.  \\)<\/li>\n<li class=\"margintop1\">\\(\\displaystyle \\int_a^b k f(x) dx = k \\int_a^b f(x) dx .  \\)<\/li>\n<li class=\"margintop1\">\ub9cc\uc57d \uc784\uc758\uc758 \\(x\\in [a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \\le g(x)\\)\uc774\uba74 \\[\\displaystyle \\int_a^b f(x) dx \\le \\int_a^b g(x) dx\\]\uc774\ub2e4.<\/li>\n<li class=\"margintop1\">\ub9cc\uc57d \\(a < c < b\\)\uc774\uba74 \\(f\\)\ub294 \\([a,\\,c]\\)\uc640 \\([c,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uba70\n\\[\\int_a^b f(x) dx = \\int_a^c f(x) dx + \\int_c^b f(x) dx \\]\n\uc774\ub2e4.\n<\/ol>\n<\/div>\n<div class=\"proof\">\n<p class=\"proofname\">\uc99d\uba85.<\/p>\n<p>\ubbf8\uc801\ubd84\ud559 \uc785\ubb38 \ub2e8\uacc4\uc5d0\uc11c \uc774 \uc815\ub9ac\uc758 \uc99d\uba85\uc740 \uc0dd\ub7b5\ud55c\ub2e4. \uc9c1\uad00\uc801\uc73c\ub85c\ub294 \uc790\uba85\ud558\uc9c0\ub9cc \uc5c4\ubc00\ud558\uac8c \uc99d\uba85\ud558\ub824\uba74 \uaf64 \uae4c\ub2e4\ub86d\ub2e4.<br \/>\n<span class=\"qed\"><\/span><\/p>\n<\/div>\n<div class=\"example\">\n<p><span class=\"example\">\ubcf4\uae30 6.1.3.<\/span> \uc55e\uc758 \ubcf4\uae30 6.1.1\uacfc 6.1.2\uc758 \uacb0\uacfc\ub97c \uc774\uc6a9\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \ub2e4\uc591\ud55c \ud568\uc218\uc758 \uc801\ubd84\uc744 \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.<\/p>\n<ol class=\"parenthesis\">\n<li>\\(\\displaystyle \\int_0^1 (x^2 +x) dx = \\int_0^1 x^2 dx + \\int_0^1 x\\,dx = \\frac{1}{3} + \\frac{1}{2} = \\frac{5}{6}. \\)<\/li>\n<li class=\"margintop1\">\\(\\displaystyle \\int_0^1 (x^2 -x) dx = \\int_0^1 x^2 dx &#8211; \\int_0^1 x\\,dx = \\frac{1}{3} &#8211; \\frac{1}{2} = -\\frac{1}{6}. \\)<\/li>\n<li class=\"margintop1\">\\(\\displaystyle \\int_0^1 (3x^2 -5x)dx = 3\\int_0^1 x^2 dx &#8211; 5\\int_0^1 x\\,dx = 3\\times \\frac{1}{3} &#8211; 5\\times\\frac{1}{2} = -\\frac{3}{2}. \\)<\/li>\n<\/ol>\n<\/div>\n<div class=\"remark\">\n<p><span class=\"remark\">\ucc38\uace0.<\/span><br \/>\n\uc774 \ucc45\uc5d0\uc11c\ub294 \ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uc778 \uacbd\uc6b0\uc5d0 \ud55c\ud574\uc11c \\(f\\)\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \uc815\uc758\ud558\uc600\ub2e4. \uc774\uc640 \uac19\uc740 \uc815\uc758\uc5d0 \ub530\ub974\uba74 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uac00 \uc544\ub2c8\ub77c\uba74 \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ub530\uc9c0\uc9c0 \uc54a\ub294 \uac83\uc774 \uc633\ub2e4.<\/p>\n<p>\ud558\uc9c0\ub9cc \uc815\uc801\ubd84\uc744 \uc815\uc758\ud558\ub294 \ub2e4\ub978 \ubc29\ubc95(\ub9ac\ub9cc \ud569\uc758 \uadf9\ud55c)\uc5d0\uc11c\ub294 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uac00 \uc544\ub2cc \uacbd\uc6b0\uc5d0\ub3c4 \\(f\\)\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ub530\uc9c4\ub2e4. (\uadf8\ub9ac\uace0 \uadf8 \uacb0\uacfc\ub294 \u201c\uc801\ubd84 \uac00\ub2a5\ud558\uc9c0 \uc54a\ub2e4\u201d\uc774\ub2e4.)<\/p>\n<p>\uadf8\ub7ec\ubbc0\ub85c \uc6b0\ub9ac\ub294 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uac00 \uc544\ub2cc \uacbd\uc6b0 \u201c\\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ub530\uc9c0\uc9c0 \uc54a\ub294\ub2e4\u201d\ub77c\uace0 \ud558\uc9c0 \uc54a\uace0 \u201c\\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uac00 \uc801\ubd84 \uac00\ub2a5\ud558\uc9c0 \uc54a\ub2e4\u201d\ub77c\uace0 \ud558\uae30\ub85c \uc57d\uc18d\ud558\uc790.<br \/>\n<span class=\"qee\"><\/span><\/p>\n<\/div>\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=\"..\/complex-differentiation\">\ubcf5\uc18c\ud568\uc218\uc758 \ubbf8\ubd84<\/a><\/li -->\n<li class=\"itc_li_prev\">\uc55e\uc758 \uae00 : <a href=\"..\/antiderivatives\">\ubd80\uc815\uc801\ubd84<\/a><\/li>\n<li class=\"itc_li_next\">\ub2e4\uc74c \uae00 : <a href=\"..\/fundamental-theorem-of-calculus\">\ubbf8\uc801\ubd84\uc758 \uae30\ubcf8\uc815\ub9ac<\/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 1\uc808\uc758 \ub0b4\uc6a9\uc785\ub2c8\ub2e4.&nbsp; (\ubbf8\uc801\ubd84\ud559 \uccab\uac78\uc74c \ucc28\ub840 \ubcf4\uae30) \uc815\uc801\ubd84\uc758 \uc815\uc758 \\(a < b\\)\uc774\uace0 \\(f\\)\uac00 \\(I=[a,\\,b]\\)\uc5d0\uc11c \uc815\uc758\ub41c \uc720\uacc4\uc778 \uc2e4\uc22b\uac12 \ud568\uc218\ub77c\uace0 \ud558\uc790. \\(f\\)\uac00 \\(I\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba70 \uc784\uc758\uc758 \\(x\\in I\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \\ge 0\\)\uc774\ub77c\uace0 \uac00\uc815\ud558\uc790. \uadf8\ub9ac\uace0 \uace1\uc120 \\(y=f(x)\\)\uc640 \\(x\\)\ucd95, \ub450 \uc9c1\uc120 \\(x=a,\\) \\(x=b\\)\ub85c \ub458\ub7ec\uc2f8\uc778 \ubd80\ubd84\uc758 \ub113\uc774\ub97c \\(A\\)\ub77c\uace0 \ud558\uc790. \uad6c\uac04 \\(I\\)\uc5d0\uc11c \ub2e4\uc74c\uacfc \uac19\uc740 \\((n+1)\\)\uac1c\uc758 \uc810 \\(x_i\\)\ub97c \ud0dd\ud558\uc790. \\(a = x_0 < x_1 < x_2 < \\cdots&hellip;\n<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":6620,"menu_order":701,"comment_status":"closed","ping_status":"closed","template":"","meta":{"_lmt_disableupdate":"","_lmt_disable":"","footnotes":""},"class_list":["post-6733","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6733","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=6733"}],"version-history":[{"count":30,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6733\/revisions"}],"predecessor-version":[{"id":8377,"href":"https:\/\/sasamath.com\/blog\/wp-json\/wp\/v2\/pages\/6733\/revisions\/8377"}],"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=6733"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}