{"id":9488,"date":"2025-10-20T18:56:00","date_gmt":"2025-10-20T09:56:00","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9488"},"modified":"2025-10-21T16:08:36","modified_gmt":"2025-10-21T07:08:36","slug":"ch06-the-riemann-integral","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-analysis\/ch06-the-riemann-integral\/","title":{"rendered":"\uc77c\ubcc0\uc218 \ud568\uc218\uc758 \uc801\ubd84"},"content":{"rendered":"
<\/p>\n

<\/p>\n

\uc774 \uc7a5\uc5d0\uc11c\ub294 \ub9ac\ub9cc \uc801\ubd84\uc758 \uc5c4\ubc00\ud55c \uc815\uc758\uc640 \uadf8 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uc801\ubd84 \uac00\ub2a5\uc131\uc758 \uc870\uac74, \ubbf8\uc801\ubd84\uc758 \uae30\ubcf8\uc815\ub9ac, \uadf8\ub9ac\uace0 \uc774\uc0c1\uc801\ubd84\uae4c\uc9c0 \uc0b4\ud3b4\ubcf8\ub2e4.<\/p>\n

\ub9ac\ub9cc \uc801\ubd84\uc758 \uc815\uc758<\/h3>\n

\uad6c\uac04 \\([a,\\, b]\\)\uc758 \uc720\ud55c\ubd80\ubd84\uc9d1\ud569 \\(P=\\left\\{ x_0 ,\\, x_1 ,\\, x_2 ,\\, \\cdots ,\\, x_n \\right\\}\\)\uc774
\n\\[a = x_0 < x_1 < x_2 < \\cdots < x_n = b\\]\n\ub97c \ub9cc\uc871\uc2dc\ud0ac \ub54c, \\(P\\)\ub97c \\([a,\\,b]\\)\uc758 \ubd84\ud560<\/span>(partition)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub610\ud55c \uac01 \uad6c\uac04
\n\\[[x_0 ,\\, x_1 ],\\, [x_1 ,\\, x_2] ,\\, \\cdots ,\\, [x_{n-1} ,\\, x_n ]\\]
\n\uc744 \\(P\\)\uc5d0 \uc758\ud558\uc5ec \ub9cc\ub4e4\uc5b4\uc9c4 \uc18c\uad6c\uac04<\/span> \ub610\ub294 \uc131\ubd84\uad6c\uac04<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc18c\uad6c\uac04\uc758 \uae38\uc774 \uc911\uc5d0\uc11c \uac00\uc7a5 \ud070 \uac12\uc744 \\(P\\)\uc758 \ub178\ub984<\/span>(norm)\uc774\ub77c\uace0 \ubd80\ub974\uba70, \\(\\lVert P \\rVert\\)\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc989
\n\\[\\|P\\| = \\max_{1 \\leq i \\leq n} (x_i – x_{i-1}) .\\]
\n\ub9cc\uc57d \\(P\\)\uc640 \\(Q\\)\uac00 \\([a,\\,b]\\)\uc758 \ubd84\ud560\uc774\uace0 \\(P\\subseteq Q\\)\uc774\uba74, \\(Q\\)\ub97c \\(P\\)\uc758 \uc138\ub828\ubd84\ud560<\/span>(refinement)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ub610\ud55c \\(P\\), \\(Q\\), \\(R\\)\uc774 \\([a,\\,b]\\)\uc758 \ubd84\ud560\uc774\uace0 \\(P\\cup Q \\subseteq R\\)\uc774\uba74 \\(R\\)\uc744 \\(P\\)\uc640 \\(Q\\)\uc758 \uacf5\ud1b5\uc138\ub828\ubd84\ud560<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

\uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uc778 \ud568\uc218 \\(f: [a,\\, b] \\to \\mathbb{R}\\)\uacfc \\([a,\\,b]\\)\uc758 \ubd84\ud560 \\(P\\)\uc5d0 \ub300\ud558\uc5ec
\n\\[\\begin{aligned}
\nm_i &= \\inf\\{f(x) \\mid x_{i-1} \\leq x \\leq x_i\\} ,\\\\[6pt]
\nM_i &= \\sup\\{f(x) \\mid x_{i-1} \\leq x \\leq x_i\\}
\n\\end{aligned}\\]
\n\ub77c\uace0 \ud558\uc790. \uc774\ub54c \\(P\\)\uc5d0 \ub300\ud55c \\(f\\)\uc758 \ub9ac\ub9cc \uc0c1\ud569<\/span>(upper Riemann sum)\uacfc \ub9ac\ub9cc \ud558\ud569<\/span>(lower Riemann sum)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
\n\\[U(f,\\, P) = \\sum_{i=1}^{n} M_i(x_i – x_{i-1}), \\quad L(f,\\, P) = \\sum_{i=1}^{n} m_i(x_i – x_{i-1}).\\]
\n\ud568\uc218 \\(f\\)\uc640 \uad6c\uac04 \\([a,\\,b]\\)\uac00 \uace0\uc815\ub418\uc5b4 \uc788\uc744 \ub54c, \uc0c1\ud569\uc758 \uac12\uacfc \ud558\ud569\uc758 \uac12\uc740 \ubd84\ud560 \\(P\\)\uc5d0 \ub530\ub77c \ub2ec\ub77c\uc9c8 \uc218 \uc788\ub2e4.<\/p>\n

\n

\ubb38\uc81c 6.1.<\/span>
\n\ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uc774\uace0, \\(P\\)\uc640 \\(Q\\)\uac00 \\([a,\\,b]\\)\uc758 \ubd84\ud560\uc774\ub77c\uace0 \ud558\uc790. \ub2e4\uc74c\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n

    \n
  1. \\(L(f,\\,P)\\le U(f,\\,P)\\)<\/li>\n
  2. \\(Q\\)\uac00 \\(P\\)\uc758 \uc138\ub828\ubd84\ud560\uc77c \ub54c \\(L(f,\\,P)\\le L(f,\\,Q)\\)\uc774\uace0 \\(U(f,\\,P)\\ge U(f,\\,Q)\\)\uc774\ub2e4.<\/li>\n
  3. \\(L(f,\\,P)\\le U(f,\\,Q)\\)<\/li>\n<\/ol>\n<\/div>\n

    \ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uc774\uace0 \\(P\\)\uac00 \\([a,\\,b]\\)\uc758 \ubd84\ud560\uc77c \ub54c \\(P\\)\uc5d0 \ub300\ud55c \\(f\\)\uc758 \uc0c1\ud569\uacfc \ud558\ud569\uc740 \uac01\uac01 \uc720\uacc4\uc774\ub2e4. \ud2b9\ud788 \\([a,\\,b]\\)\uc758 \uc784\uc758\uc758 \ubd84\ud560 \\(P\\), \\(Q\\)\uc5d0 \ub300\ud558\uc5ec \\(L(f,\\,P) \\le U(f,\\,Q)\\)\uc774\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \uc0c1\uc801\ubd84<\/span>(upper integral)\uacfc \ud558\uc801\ubd84<\/span>(lower integral)\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
    \n\\[\\begin{aligned}
    \n\\overline{\\int_a^b} f &= \\inf \\left\\{ U(f,\\, P) \\mid P \\text{ is a partition of }[a,\\,b]\\right\\}, \\\\[6pt]
    \n\\underline{\\int_a^b} f &= \\sup \\left\\{ L(f,\\, P)\\mid P \\text{ is a partition of }[a,\\,b]\\right\\}.
    \n\\end{aligned}\\]
    \n\ud568\uc218 \\(f\\)\uac00 \\([a,\\, b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5<\/span>\ud558\ub2e4(Riemann integrable)\ub294 \uac83\uc740 \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uac00 \uc720\uacc4\uc774\uace0 \\(f\\)\uc758 \uc0c1\uc801\ubd84\uacfc \ud558\uc801\ubd84\uc774 \uac19\uc740 \uac83\uc774\ub2e4. \uc774 \uacf5\ud1b5\uac12\uc744
    \n\\[\\int_a^b f(x) dx\\]
    \n\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc774 \uac12\uc744 \uac04\ub2e8\ud788 \\(\\int_a^b f\\)\ub85c \ub098\ud0c0\ub0b4\uae30\ub3c4 \ud55c\ub2e4.<\/p>\n

    \ub9ac\ub9cc \uc801\ubd84\uc758 \uc815\uc758\uc640 \uc0c1\ud55c\uc758 \uc131\uc9c8\uc5d0 \uc758\ud558\uc5ec \ub2e4\uc74c \uc815\ub9ac\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 6.1. (\uc801\ubd84 \uac00\ub2a5\uc131\uc5d0 \ub300\ud55c \ub9ac\ub9cc \ud310\uc815\ubc95)<\/span><\/p>\n

    \ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740, \uc784\uc758\uc758 \\(\\varepsilon > 0\\)\uc5d0 \ub300\ud574 \ubd84\ud560 \\(P\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(U(f,\\, P) – L(f,\\, P) < \\varepsilon\\)\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

    \n

    \ubb38\uc81c 6.2.<\/span>
    \n\uc801\ubd84\uc758 \uc815\uc758\ub97c \uc0ac\uc6a9\ud558\uc5ec \uad6c\uac04 \\([0,\\, 1]\\)\uc5d0\uc11c \ud568\uc218 \\(f(x) = x^2\\)\uc758 \uc801\ubd84\uc744 \uacc4\uc0b0\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

    \n

    \ubb38\uc81c 6.3.<\/span>
    \n\ub514\ub9ac\ud074\ub808 \ud568\uc218\uac00 \\([0,\\, 1]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uc9c0 \uc54a\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

    \n

    \ubb38\uc81c 6.4.<\/span>
    \n\uc815\ub9ac 6.1\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

    \uc0c1\ud569\uacfc \ud558\ud569 \ub300\uc2e0 \ub9ac\ub9cc\ud569\uc744 \uc0ac\uc6a9\ud558\uc5ec \ub9ac\ub9cc \uc801\ubd84\uc744 \uc815\uc758\ud560 \uc218\ub3c4 \uc788\ub2e4. \uac01 \uc18c\uad6c\uac04 \\([x_{i-1},\\, x_i]\\)\uc5d0\uc11c \uc810 \\(t_i\\)\ub97c \uc120\ud0dd\ud558\uc5ec \ub9cc\ub4e0 \ud569
    \n\\[S(f,\\, P,\\, \\{t_i\\}) = \\sum_{i=1}^{n} f(t_i)(x_i – x_{i-1})\\]
    \n\uc744 \\(P\\)\uc640 \\(\\left\\{ t_i \\right\\}\\)\uc5d0 \ub300\ud55c \\(f\\)\uc758 \ub9ac\ub9cc \ud569<\/span>(Riemann sum)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \\(\\lVert P \\rVert \\rightarrow 0\\)\uc77c \ub54c \\(f\\)\uc758 \ub9ac\ub9cc \ud569\uc774 \uac12 \\(I\\)\uc5d0 \uc218\ub834\ud55c\ub2e4\ub294 \uac83\uc740, \uc784\uc758\uc758 \\(\\varepsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(\\delta > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(\\lVert P \\rVert < \\delta\\)\uc778 \uc784\uc758\uc758 \ubd84\ud560 \\(P\\)\uc640 \\(P\\)\uc758 \uac01 \uc18c\uad6c\uac04\uc5d0\uc11c \ud0dd\ud55c \uc810\uc73c\ub85c \uc774\ub8e8\uc5b4\uc9c4 \uc784\uc758\uc758 \uc720\ud55c\uc218\uc5f4 \\(\\left\\{ t_i \\right\\}\\)\uc5d0 \ub300\ud558\uc5ec \\(\\lvert S(f,\\,P,\\,\\left\\{ t_i \\right\\}) - I \\rvert < \\varepsilon\\)\uc774 \uc131\ub9bd\ud558\ub294 \uac83\uc744 \ub73b\ud55c\ub2e4. \uc774\uac83\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.\n\\[\\lim_{\\lVert P \\rVert \\rightarrow 0} S(f,\\,P,\\,\\left\\{ t_i \\right\\}) = I.\\tag{6.1}\\]<\/p>\n

    \ub9ac\ub9cc\uc740 \uc801\ubd84\uc744 \ub9ac\ub9cc\ud569\uc758 \uadf9\ud55c\uc73c\ub85c \uc815\uc758\ud588\ub2e4. \uc0c1\ud569\uacfc \ud558\ud569\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc815\uc758\ud55c \uc801\ubd84\uc740 \ubcf8\ub798 \ub2e4\ub974\ubd80(Darboux)\uc758 \uc801\ubd84\uc774\ub2e4. \uadf8\ub7ec\ub098 \uc720\uacc4\uc778 \ud568\uc218\uc5d0 \ub300\ud558\uc5ec \uc774 \ub450 \uc815\uc758\ub294 \uc11c\ub85c \ub3d9\uce58\uc774\ub2e4. \uc989, \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uc778 \ud568\uc218 \\(f\\)\uc5d0 \ub300\ud558\uc5ec, \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\uc801\ubd84\uacfc \ud558\uc801\ubd84\uc774 \uc77c\uce58\ud558\uae30 \uc704\ud55c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \ub9ac\ub9cc\ud569\uc774 \ud558\ub098\uc758 \uc2e4\uc218\uc5d0 \uc218\ub834\ud558\ub294 \uac83\uc774\ub2e4. \uc774\ub54c \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc0c1\uc801\ubd84\uc758 \uac12\uacfc \\([a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \ub9ac\ub9cc\ud569\uc758 \uadf9\ud55c\uac12\uc740 \uc77c\uce58\ud55c\ub2e4.<\/p>\n

    \n

    \ubb38\uc81c 6.5.<\/span>
    \n\ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uace0 \ud568\uc218 \\(g\\)\uac00 \\([c,\\,d]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba70 \\(f([a,\\,b])\\subseteq [c,\\,d]\\)\uc77c \ub54c, \ud569\uc131\ud568\uc218 \\(g\\circ f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

    \n

    \ubb38\uc81c 6.6.<\/span>
    \n\ud568\uc218 \\(f\\)\uc640 \\(g\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uba74 \\(f+g\\)\uc640 \\(fg\\)\ub3c4 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

    \n

    \ubb38\uc81c 6.7.<\/span>
    \n\uc801\ubd84\uc5d0 \ub300\ud55c \ub9ac\ub9cc\uc758 \uc815\uc758\uc640 \ub2e4\ub974\ubd80\uc758 \uc815\uc758\uac00 \uc11c\ub85c \ub3d9\uce58\uc784\uc744 \uc99d\uba85\ud558\ub294 \uacfc\uc815\uc744 \uc870\uc0ac\ud558\uc2dc\uc624. \uc989 \uc815\ub9ac 6.1\uacfc \ub9ac\ub9cc\ud569\uc758 \uadf9\ud55c (6.1)\uc774 \uc218\ub834\ud558\ub294 \uac83\uc774 \uc11c\ub85c \ub3d9\uce58\uc784\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

    \n

    \ubb38\uc81c 6.8.<\/span>
    \n\ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \uc774\ub54c \ubb38\uc81c 6.7\uc758 \uacb0\uacfc\ub97c \uc0ac\uc6a9\ud558\uc5ec \ub2e4\uc74c\uc774 \uc131\ub9bd\ud568\uc744 \ubcf4\uc774\uc2dc\uc624.
    \n\\[\\int_a^b f(x) dx = \\lim_{n\\rightarrow\\infty}\\frac{b-a}{n}\\sum_{k=1}^{n}f\\left(a+\\frac{b-a}{n}k\\right).\\]<\/p>\n<\/div>\n

    \uc801\ubd84 \uac00\ub2a5\uc131 \uc870\uac74<\/h3>\n

    \uc801\ubd84\uc744 \uc815\uc758\ud55c \ub4a4\uc5d0\ub294 \ub2e4\uc74c\uacfc \uac19\uc740 \ub450 \uac00\uc9c0 \uc758\ubb38\uc774 \uc0dd\uae34\ub2e4.<\/p>\n

      \n
    • \uc5b4\ub5a0\ud55c \ud568\uc218\uac00 \uc801\ubd84 \uac00\ub2a5\ud55c\uac00?<\/li>\n
    • \uc801\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uac00 \uc788\uc744 \ub54c, \uadf8 \ud568\uc218\uc758 \uc801\ubd84\uac12\uc744 \uc5b4\ub5bb\uac8c \uad6c\ud558\ub294\uac00?<\/li>\n<\/ul>\n
      \n

      \uc815\ub9ac 6.2. (\uc5f0\uc18d\ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131)<\/span><\/p>\n

      \ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74, \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/p>\n

      \\(f\\)\uac00 \\([a,\\, b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74, \uc774 \uad6c\uac04\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\ub2e4. \\(\\varepsilon > 0\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uade0\ub4f1\uc5f0\uc18d\uc131\uc758 \uc815\uc758\uc5d0 \uc758\ud558\uc5ec, \\(\\delta > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(|x – y| < \\delta\\)\uc778 \uc784\uc758\uc758 \\(x,\\,y\\in [a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(|f(x) - f(y)| < \\frac{\\varepsilon}{b-a}\\)\uc774 \uc131\ub9bd\ud55c\ub2e4. \\(\\|P\\| < \\delta\\)\uc778 \ubd84\ud560 \\(P\\)\ub97c \ud0dd\ud558\uba74 \\(M_i - m_i < \\frac{\\varepsilon}{b-a}\\)\uc774\ub2e4. \ub530\ub77c\uc11c \\(U(f,\\, P) - L(f,\\, P) < \\varepsilon\\)\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

      \n

      \uc815\ub9ac 6.3. (\ub2e8\uc870\ud568\uc218\uc758 \uc801\ubd84 \uac00\ub2a5\uc131)<\/span><\/p>\n

      \ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ub2e8\uc870\uc774\uba74, \\(f\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/p>\n

      \ub9cc\uc57d \\(f(a)=f(b)\\)\uc774\uba74 \\(f\\)\ub294 \uc0c1\uc218\ud568\uc218\uc774\ubbc0\ub85c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \uc774\uc81c \\(f(a)\\ne f(b)\\)\ub77c\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\uace0 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ub2e8\uc870\uc99d\uac00\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790.<\/p>\n

      \\(\\varepsilon > 0\\)\uc774 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \uadf8\ub9ac\uace0 \\(\\lVert P \\rVert < \\frac{\\varepsilon}{f(b)-f(a)}\\)\uc778 \ubd84\ud560 \\(P\\)\ub97c \ud0dd\ud55c\ub2e4. \uadf8\ub7ec\uba74 \\(U(f,\\,P) - L(f,\\,P) \\le \\lVert P \\rVert (f(b)-f(a)) < \\varepsilon\\)\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

      \ud568\uc218 \\(f\\)\uac00 \ubd88\uc5f0\uc18d\uc778 \uc810\uc774 \uad6c\uac04 \\([a,\\,b]\\)\uc5d0 \uc5b4\ub5bb\uac8c \ubd84\ud3ec\ud574 \uc788\ub294\uc9c0\uc5d0 \ub530\ub77c \\(f\\)\uc758 \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \ud310\ubcc4\ud560 \uc218 \uc788\ub2e4. \uc774\uc640 \uad00\ub828\ub41c \uc815\ub9ac\ub97c \uc9c4\uc220\ud558\uae30 \uc704\ud574\uc11c\ub294 \ub974\ubca0\uadf8 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc758 \uac1c\ub150\uc774 \ud544\uc694\ud558\ub2e4.<\/p>\n

      \uc9d1\ud569 \\(E \\subseteq \\mathbb{R}\\)\uc774 \ub974\ubca0\uadf8 \uce21\ub3c4 0<\/span>(Lebesgue measure zero)\uc778 \uc9d1\ud569\uc774\ub77c\ub294 \uac83\uc740, \uc784\uc758\uc758 \\(\\varepsilon > 0\\)\uc5d0 \ub300\ud558\uc5ec \uac00\uc0b0 \uac1c\uc758 \uad6c\uac04 \\(\\{I_n\\}\\)\uc774 \uc874\uc7ac\ud558\uc5ec \\(E \\subseteq \\bigcup_{n=1}^{\\infty} I_n\\)\uc774\uace0 \\(\\sum_{n=1}^{\\infty} |I_n| < \\varepsilon\\)\uc778 \uac83\uc774\ub2e4.<\/p>\n

        \n
      • \uc720\ud55c\uc9d1\ud569\uc758 \uce21\ub3c4\ub294 \\(0\\)\uc774\ub2e4.<\/li>\n
      • \uac00\uc0b0\uc9d1\ud569\uc758 \uce21\ub3c4\ub294 \\(0\\)\uc774\ub2e4.<\/li>\n
      • \uce78\ud1a0\uc5b4\uc758 \uc9d1\ud569\uc740 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\uc9c0\ub9cc \uce21\ub3c4 \\(0\\)\uc774\ub2e4.<\/li>\n
      • \uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04 \\([a,\\,b]\\)\ub294 \ube44\uac00\uc0b0\uc9d1\ud569\uc774\uace0, \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc774 \uc544\ub2c8\ub2e4.<\/li>\n<\/ul>\n
        \n

        \ubb38\uc81c 6.9.<\/span>
        \n\uac00\uc0b0\uc9d1\ud569\uc774 \ub974\ubca0\uadf8 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

        \n

        \ubb38\uc81c 6.10.<\/span>
        \n\uac00\uc0b0 \uac1c\uc758 \uc9d1\ud569 \\(A_1\\), \\(A_2\\), \\(A_3\\), \\(\\cdots\\)\uac00 \ubaa8\ub450 \ub974\ubca0\uadf8 \uce21\ub3c4 \\(0\\)\uc774\uba74, \ud569\uc9d1\ud569
        \n\\[A_1 \\cup A_2 \\cup A_3 \\cup \\cdots \\]
        \n\ub3c4 \ub974\ubca0\uadf8 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc784\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

        \n

        \ubb38\uc81c 6.11.<\/span>
        \n\uae38\uc774\uac00 \uc591\uc218\uc778 \uad6c\uac04\uc774 \ub974\ubca0\uadf8 \uce21\ub3c4 \\(0\\)\uc778 \uc9d1\ud569\uc774 \uc544\ub2d8\uc744 \ubcf4\uc774\uc2dc\uc624.<\/p>\n<\/div>\n

        \n

        \uc815\ub9ac 6.4. (\uc801\ubd84 \uac00\ub2a5\uc131\uc5d0 \ub300\ud55c \ub974\ubca0\uadf8 \uc815\ub9ac)<\/span><\/p>\n

        \uad6c\uac04 \\([a,\\,b]\\)\uc5d0\uc11c \uc720\uacc4\uc778 \ud568\uc218 \\(f\\)\uac00 \uc774 \uad6c\uac04\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 \uc9d1\ud569\uc774 \ub974\ubca0\uadf8 \uce21\ub3c4 0\uc778 \uc9d1\ud569\uc778 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

        \n

        \uc99d\uba85 \uac1c\uc694<\/span>
        \n\ub2e4\uc74c\uacfc \uac19\uc740 \uc21c\uc11c\ub85c \uc99d\uba85\ud55c\ub2e4.<\/p>\n

        (\\(\\Rightarrow\\)) \uba3c\uc800 \\(f\\)\uac00 \uc801\ubd84 \uac00\ub2a5\ud558\uba74 \ubd88\uc5f0\uc18d\uc810\uc758 \uc9d1\ud569\uc774 \uce21\ub3c4 0\uc784\uc744 \ubcf4\uc774\uc790.<\/p>\n

        \n
          \n
        1. \uc810 \\(x\\)\uc5d0\uc11c \\(f\\)\uc758 \uc9c4\ub3d9<\/span>(oscillation)\uc744 \\(\\omega(x) = \\lim_{\\delta \\to 0+} \\sup\\{|f(y) – f(z)| \\mid |y-x| < \\delta,\\, |z-x| < \\delta\\}\\)\ub85c \uc815\uc758\ud55c\ub2e4.<\/li>\n
        2. \\(f\\)\uac00 \\(x\\)\uc5d0\uc11c \uc5f0\uc18d\uc77c \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \\(\\omega(x) = 0\\)\uc778 \uac83\uc774\ub2e4.<\/li>\n
        3. \ubd88\uc5f0\uc18d\uc810\uc758 \uc9d1\ud569\uc744 \\(D = \\{x \\in [a,\\,b] \\mid \\omega(x) > 0\\} = \\bigcup_{n=1}^{\\infty} D_n\\)\uc73c\ub85c \ub098\ud0c0\ub0b8\ub2e4. \uc5ec\uae30\uc11c \\(D_n = \\{x \\mid \\omega(x) \\geq 1\/n\\}\\)\uc774\ub2e4.<\/li>\n
        4. \\(f\\)\uac00 \uc801\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c, \uc784\uc758\uc758 \\(\\varepsilon > 0\\)\uc5d0 \ub300\ud574 \ubd84\ud560 \\(P\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(U(f,\\, P) – L(f,\\, P) < \\varepsilon\/n\\)\uc774\ub2e4.<\/li>\n
        5. \\(D_n\\)\uc5d0 \uc18d\ud558\ub294 \uc810\uc744 \ud3ec\ud568\ud558\ub294 \uc18c\uad6c\uac04\ub4e4\uc758 \uae38\uc774\uc758 \ud569\uc774 \\(\\varepsilon\\) \ubbf8\ub9cc\uc784\uc744 \ubcf4\uc77c \uc218 \uc788\ub2e4.<\/li>\n
        6. \ub530\ub77c\uc11c \uac01 \\(D_n\\)\uc774 \uce21\ub3c4 0\uc774\uace0, \uac00\uc0b0 \ud569\uc9d1\ud569 \\(D\\)\ub3c4 \uce21\ub3c4 0\uc774\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

          (\\(\\Leftarrow\\)) \ubd88\uc5f0\uc18d\uc810\uc758 \uc9d1\ud569\uc774 \uce21\ub3c4 0\uc774\uba74 \\(f\\)\uac00 \uc801\ubd84 \uac00\ub2a5\ud568\uc744 \ubcf4\uc774\uc790.<\/p>\n

          \n
            \n
          1. \ubd88\uc5f0\uc18d\uc810\uc758 \uc9d1\ud569 \\(D\\)\uac00 \uce21\ub3c4 0\uc774\ubbc0\ub85c, \uc784\uc758\uc758 \\(\\varepsilon > 0\\)\uc5d0 \ub300\ud574 \\(D\\)\ub97c \ub36e\ub294 \uac00\uc0b0 \uac1c\uc758 \uc5f4\ub9b0\uad6c\uac04\ub4e4\uc758 \uae38\uc774\uc758 \ud569\uc744 \\(\\varepsilon\/2M\\) \ubbf8\ub9cc\uc73c\ub85c \ub9cc\ub4e4 \uc218 \uc788\ub2e4. (\uc5ec\uae30\uc11c \\(|f| \\leq M\\))<\/li>\n
          2. \\(f\\)\uac00 \uc5f0\uc18d\uc778 \uc810\uc5d0\uc11c\ub294 \uc9c4\ub3d9\uc774 \\(0\\)\uc774\ubbc0\ub85c, \uc801\uc808\ud55c \ubd84\ud560\uc744 \uc7a1\uc544 \uc5f0\uc18d\uc810\uc744 \ud3ec\ud568\ud558\ub294 \uc18c\uad6c\uac04\uc5d0\uc11c\uc758 \uc0c1\ud569\uacfc \ud558\ud569\uc758 \ucc28\uac00 \\(\\varepsilon\/2\\) \ubbf8\ub9cc\uc774 \ub418\ub3c4\ub85d \ub9cc\ub4e4 \uc218 \uc788\ub2e4.<\/li>\n
          3. \ub610\ud55c \ubd88\uc5f0\uc18d\uc810\uc744 \ud3ec\ud568\ud558\ub294 \uc18c\uad6c\uac04\uc5d0\uc11c\uc758 \uc0c1\ud569\uacfc \ud558\ud569\uc758 \ucc28\uac00 \\(2M \\cdot \\varepsilon\/2M = \\varepsilon\/2\\) \uc774\ud558\uac00 \ub418\ub3c4\ub85d \ubd84\ud560\uc744 \uad6c\uc131\ud55c\ub2e4.<\/li>\n
          4. \ub530\ub77c\uc11c \\(U(f,\\, P) – L(f,\\, P) < \\varepsilon\\)\uc778 \ubd84\ud560 \\(P\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.<\/span><\/li>\n<\/ol>\n<\/div>\n<\/div>\n

            \ub974\ubca0\uadf8 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud558\uc5ec \uc801\ubd84 \uac00\ub2a5\uc131\uc744 \uc27d\uac8c \ud310\ubcc4\ud560 \uc218 \uc788\ub294 \ud568\uc218\uc758 \uc608\ub294 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n

              \n
            • \uc720\ud55c \uac1c\uc758 \uc810\uc5d0\uc11c\ub9cc \ubd88\uc5f0\uc18d\uc778 \ud568\uc218\ub294 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/li>\n
            • \uac00\uc0b0 \uac1c\uc758 \uc810\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc778 \ud568\uc218\ub294 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/li>\n
            • \uc720\ub9ac\uc218\uc778 \uc810\uc5d0\uc11c \ud568\uc22b\uac12\uc774 \\(1\\)\uc774\uace0 \ubb34\ub9ac\uc218\uc778 \uc810\uc5d0\uc11c \ud568\uc22b\uac12\uc774 \\(0\\)\uc778 \ud568\uc218\ub97c \ub514\ub9ac\ud074\ub808 \ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \ub514\ub9ac\ud074\ub808 \ud568\uc218\ub294 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ubd88\uc5f0\uc18d\uc774\uba70, \uc801\ubd84 \ubd88\uac00\ub2a5\ud558\ub2e4.<\/li>\n<\/ul>\n
              \n

              \ubb38\uc81c 6.12.<\/span>
              \n\ud568\uc218 \\(f\\)\uac00 \\([0,\\,1]\\)\uc5d0\uc11c \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ub418\uc5b4 \uc788\ub2e4.
              \n\\[f(x)=
              \n\\begin{cases}
              \n\\frac{1}{q} &\\,\\,\\,\\text{if }\\,x\\in [0,\\,1]\\cap\\mathbb{Q},\\, x=\\frac{p}{q},\\, p\\in\\mathbb{Z},\\,q\\in\\mathbb{N},\\,\\operatorname{gcd}(p,\\,q)=1. \\\\[6pt]
              \n0 &\\,\\,\\,\\text{if }\\,x\\in [0,\\,1]\\setminus\\mathbb{Q}.
              \n\\end{cases}
              \n\\]
              \n\uc774 \ud568\uc218\ub97c \ud1a0\ub9e4 \ud568\uc218<\/span>(Thomae function) \ub610\ub294 \ud31d\ucf58 \ud568\uc218<\/span>\ub77c\uace0 \ubd80\ub978\ub2e4. \ub2e4\uc74c\uc744 \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n

                \n
              1. \ud568\uc218 \\(f\\)\uac00 \\([0,\\,1]\\)\uc758 \uc720\ub9ac\uc218\uc778 \uc810\uc5d0\uc11c\ub294 \ubd88\uc5f0\uc18d\uc774\uace0 \ubb34\ub9ac\uc218\uc778 \uc810\uc5d0\uc11c\ub294 \uc5f0\uc18d\uc774\ub2e4.<\/li>\n
              2. \ud568\uc218 \\(f\\)\ub294 \\([0,\\,1]\\)\uc758 \ubaa8\ub4e0 \uc810\uc5d0\uc11c \ubbf8\ubd84 \ubd88\uac00\ub2a5\ud558\ub2e4.<\/li>\n
              3. \ud568\uc218 \\(f\\)\ub294 \\([0,\\,1]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/li>\n<\/ol>\n<\/div>\n

                \uc801\ubd84\uc758 \uc131\uc9c8<\/h3>\n

                \uc801\ubd84\uc758 \uae30\ubcf8 \uc131\uc9c8\uc744 \uc0b4\ud3b4\ubcf4\uc790. \\(a < b\\)\uc774\uace0 \ud568\uc218 \\(f\\)\uc640 \\(g\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790.<\/p>\n

                  \n
                • \uc120\ud615\uc131: \\(\\alpha\\)\uc640 \\(\\beta\\)\uac00 \uc2e4\uc218\uc778 \uc0c1\uc218\uc77c \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
                  \n\\[\\int_a^b (\\alpha f(x) + \\beta g(x))dx = \\alpha \\int_a^b f(x)dx + \\beta \\int_a^b g(x)dx.\\tag{6.2}\\]<\/li>\n
                • \ub2e8\uc870\uc131: \uc784\uc758\uc758 \\(x\\in[a,\\,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(f(x) \\leq g(x)\\)\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
                  \n\\[\\int_a^b f(x)dx \\leq \\int_a^b g(x)dx.\\tag{6.3}\\]<\/li>\n
                • \uc808\ub313\uac12: \\(|f|\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uace0 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
                  \n\\[\\left|\\int_a^b f(x)dx \\right| \\leq \\int_a^b |f(x)|dx.\\tag{6.4}\\]<\/li>\n
                • \uc801\ubd84 \uad6c\uac04\uc758 \uac00\ubc95\uc131: \\(a < c < b\\)\uc77c \ub54c \\(f\\)\ub294 \\([a,\\,c]\\)\uc640 \\([c,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uace0 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.\n\\[\\int_a^c f(x)dx + \\int_c^b f(x)dx = \\int_a^b f(x)dx.\\tag{6.5}\\]<\/li>\n<\/ul>\n
                  \n

                  \ubb38\uc81c 6.13.<\/span>
                  \n\uc801\ubd84\uc758 \uc131\uc9c8 (6.2), (6.3), (6.4), (6.5)\ub97c \uc99d\uba85\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

                  \uc784\uc758\uc758 \uc2e4\uc218 \\(a\\)\uc5d0 \ub300\ud558\uc5ec \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
                  \n\\[\\int_a^a f(x)dx =0.\\]
                  \n\ub610\ud55c \\(a>b\\)\uc778 \uacbd\uc6b0 \ub2e4\uc74c\uacfc \uac19\uc774 \uc815\uc758\ud55c\ub2e4.
                  \n\\[\\int_b^a f(x) dx = -\\int_a^b f(x) dx .\\]
                  \n\uc774\uc640 \uac19\uc774 \uc815\uc758\ud558\uba74, \ud568\uc218 \\(f\\)\uac00 \ub2eb\ud78c\uad6c\uac04 \\(I\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uace0 \\(a\\), \\(b\\), \\(c\\)\uac00 \\(I\\)\uc758 \uc810\uc77c \ub54c\ub3c4 (6.5)\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

                  \ubbf8\uc801\ubd84\uc758 \uae30\ubcf8\uc815\ub9ac<\/h3>\n

                  \ud568\uc218 \\(f\\)\uac00 \\([a,\\, b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud560 \ub54c, \uc801\ubd84\ud568\uc218<\/span>(integral function)\ub97c
                  \n\\[F(x) = \\int_a^x f(t) dt\\]
                  \n\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n

                  \n

                  \uc815\ub9ac 6.5. (\ubbf8\uc801\ubd84\uc758 \uc81c1\uae30\ubcf8\uc815\ub9ac)<\/span><\/p>\n

                  \\(f\\)\uac00 \\([a,\\, b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uba74 \\(F(x) = \\int_a^x f(t) dt\\)\ub294 \\([a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ub2e4. \ub610\ud55c \\(f\\)\uac00 \\(c\\in [a,\\,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74 \\(F'(c) = f(c)\\)\uc774\ub2e4.<\/p>\n<\/div>\n

                  \n

                  \uc99d\uba85<\/p>\n

                  \\(\\lvert f \\rvert \\le M\\)\uc774\ub77c\uace0 \ud558\uc790. \uc808\ub313\uac12\uc774 \uc791\uc740 \\(h\\)\uc5d0 \ub300\ud558\uc5ec
                  \n\\[|F(x + h) – F(x)| = \\left|\\int_x^{x+h} f(t) dt\\right| \\leq M|h|\\]
                  \n\uc774\ubbc0\ub85c \\(F\\)\ub294 \uc5f0\uc18d\uc774\ub2e4. \ub9cc\uc57d \\(f\\)\uac00 \\(c\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74, \uc784\uc758\uc758 \\(\\varepsilon > 0\\)\uc5d0 \ub300\ud574 \\(\\delta > 0\\)\uc774 \uc874\uc7ac\ud558\uc5ec, \\(|t – c| < \\delta\\)\uc77c \ub54c \\(|f(t) - f(c)| < \\varepsilon\\)\uc774\ub2e4. \\(|h| < \\delta\\)\uc77c \ub54c\n\\[\\left|\\frac{F(c + h) - F(c)}{h} - f(c)\\right| < \\varepsilon\\]\n\uc774 \uc131\ub9bd\ud55c\ub2e4. \uadf8\ub7ec\ubbc0\ub85c \\(F ' (c) = f(c)\\)\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

                  \uc704 \uc815\ub9ac\uc758 \ub530\ub984\uc815\ub9ac\ub85c\uc11c \ub2e4\uc74c\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

                  \n

                  \uc815\ub9ac 6.6. (\ubbf8\uc801\ubd84\uc758 \uc81c2\uae30\ubcf8\uc815\ub9ac)<\/span><\/p>\n

                  \ud568\uc218 \\(f\\)\uac00 \\([a,\\, b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\(F\\)\uac00 \\(f\\)\uc758 \uc6d0\uc2dc\ud568\uc218\uc774\uba74 \ub2e4\uc74c\uc774 \uc131\ub9bd\ud55c\ub2e4.
                  \n\\[\\int_a^b f(x) dx = F(b) – F(a).\\]<\/p>\n<\/div>\n

                  \n

                  \uc99d\uba85<\/p>\n

                  \\(f\\)\uac00 \\([a, b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \\(F\\)\uac00 \\(f\\)\uc758 \uc6d0\uc2dc\ud568\uc218\ub77c\uace0 \ud558\uc790. \uc989, \ubaa8\ub4e0 \\(x \\in [a, b]\\)\uc5d0 \ub300\ud574 \\(F'(x) = f(x)\\)\uc774\ub2e4.
                  \n\uc81c1\uae30\ubcf8\uc815\ub9ac\uc5d0 \uc758\ud574 \\(G(x) = \\int_a^x f(t) dt\\)\ub3c4 \\(f\\)\uc758 \uc6d0\uc2dc\ud568\uc218\uc774\ub2e4.
                  \n\ub450 \uc6d0\uc2dc\ud568\uc218 \\(F\\)\uc640 \\(G\\)\uc758 \ucc28\uc774\ub294 \uc0c1\uc218\uc774\ubbc0\ub85c, \uc5b4\ub5a4 \uc0c1\uc218 \\(C\\)\uc5d0 \ub300\ud574
                  \n\\[F(x) = G(x) + C = \\int_a^x f(t) dt + C\\]
                  \n\uc774\ub2e4. \\(x = a\\)\ub97c \ub300\uc785\ud558\uba74
                  \n\\[F(a) = \\int_a^a f(t) dt + C = 0 + C = C\\]
                  \n\uc774\ubbc0\ub85c \\(C = F(a)\\)\uc774\ub2e4. \ub530\ub77c\uc11c \ubaa8\ub4e0 \\(x \\in [a, b]\\)\uc5d0 \ub300\ud574
                  \n\\[F(x) = \\int_a^x f(t) dt + F(a)\\]
                  \n\uc774\ub2e4. \\(x = b\\)\ub97c \ub300\uc785\ud558\uba74
                  \n\\[F(b) = \\int_a^b f(t) dt + F(a)\\]
                  \n\uc774\ub2e4. \\(F(a)\\)\ub97c \uc774\ud56d\ud558\uba74 \ubc14\ub77c\ub294 \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

                  \uc704 \uc815\ub9ac\uc758 \uc6b0\ubcc0\uc744 \ub2e4\uc74c\uacfc \uac19\uc774 \ub098\ud0c0\ub0b8\ub2e4.
                  \n\\[\\Big[F(x)\\Big]_a^b = F(b) – F(a) \\quad\\text{ \ub610\ub294 }\\quad F(x) \\Big\\vert _a ^b = F(b) -F(a).\\]<\/p>\n

                  \ubbf8\uc801\ubd84\uc758 \uae30\ubcf8\uc815\ub9ac\ub85c\ubd80\ud130 \ub2e4\uc74c\uacfc \uac19\uc740 \uc801\ubd84\uc758 \uacc4\uc0b0 \uacf5\uc2dd\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

                    \n
                  • \ubd80\ubd84\uc801\ubd84\ubc95: \ud568\uc218 \\(u\\)\uc640 \\(v\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \\(C^1\\)\uc77c \ub54c
                    \n\\[\\int_a^b u \\,dv = [uv]_a^b – \\int_a^b v \\,du.\\]<\/li>\n
                  • \uce58\ud658\uc801\ubd84\ubc95: \ud568\uc218 \\(g\\)\uac00 \\([c,\\, d]\\)\uc5d0\uc11c \\([a,\\, b]\\)\ub85c\uc758 \\(C^1\\) \ud568\uc218\uc774\uace0, \ud568\uc218 \\(f\\)\uac00 \\(g([c,\\,d])\\)\uc5d0\uc11c \uc5f0\uc18d\uc77c \ub54c
                    \n\\[\\int_{g(c)}^{g(d)} f(x) \\,dx = \\int_c^d f(g(t))g'(t) \\,dt.\\]<\/li>\n<\/ul>\n
                    \n

                    \ubb38\uc81c 6.14.<\/span>
                    \n\ubd80\ubd84\uc801\ubd84\ubc95\uc744 \uc0ac\uc6a9\ud558\uc5ec \\(\\int_0^{\\pi} x \\sin x \\,dx\\)\ub97c \uacc4\uc0b0\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

                    \n

                    \ubb38\uc81c 6.15.<\/span>
                    \n\ud568\uc218 \\(f\\)\uac00 \\([a,\\,b]\\)\uc5d0\uc11c \ubbf8\ubd84 \uac00\ub2a5\ud558\uace0 \\(f’\\)\uc774 \\([a,\\,b]\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud560 \ub54c \ub2e4\uc74c\uc774 \uc131\ub9bd\ud568\uc744 \uc99d\uba85\ud558\uc2dc\uc624.
                    \n\\[\\int_{a}^{b} f’ (x) \\,dx = f(b)-f(a).\\]<\/p>\n<\/div>\n

                    \uc801\ubd84\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac<\/h3>\n

                    \ubbf8\uc801\ubd84\uc758 \uae30\ubcf8\uc815\ub9ac\ub97c \ud65c\uc6a9\ud558\uc5ec \ub04c\uc5b4\ub0bc \uc218 \uc788\ub294 \uc815\ub9ac\ub85c\uc11c \uc801\ubd84\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac\uac00 \uc788\ub2e4.<\/p>\n

                    \n

                    \uc815\ub9ac 6.7. (\uc801\ubd84\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac)<\/span><\/p>\n

                    \ud568\uc218 \\(f\\)\uac00 \\([a,\\, b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74, \ub2e4\uc74c \ub4f1\uc2dd\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac12 \\(c \\in [a,\\, b]\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.
                    \n\\[\\frac{1}{b-a}\\int_a^b f(x) dx = f(c).\\]<\/p>\n<\/div>\n

                    \n

                    \uc99d\uba85<\/p>\n

                    \\(f\\)\uac00 \\([a, b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\ubbc0\ub85c \ucd5c\ub313\uac12 \\(M\\)\uacfc \ucd5c\uc19f\uac12 \\(m\\)\uc744 \uac00\uc9c4\ub2e4. \uc989 \ubaa8\ub4e0 \\(x \\in [a, b]\\)\uc5d0 \ub300\ud574 \\(m \\leq f(x) \\leq M\\)\uc774\ub2e4. \uc801\ubd84\uc758 \ub2e8\uc870\uc131\uc5d0 \uc758\ud574
                    \n\\[m(b-a) \\leq \\int_a^b f(x) dx \\leq M(b-a)\\]
                    \n\uc774\ub2e4. \ub530\ub77c\uc11c
                    \n\\[m \\leq \\frac{1}{b-a}\\int_a^b f(x) dx \\leq M\\]
                    \n\uc774\ub2e4. \uc0ac\uc787\uac12 \uc815\ub9ac\uc5d0 \uc758\ud574 \uc5b4\ub5a4 \\(c \\in [a, b]\\)\uc5d0 \ub300\ud574
                    \n\\[f(c) = \\frac{1}{b-a}\\int_a^b f(x) dx\\]\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

                    \n

                    \uc815\ub9ac 6.8. (\uc81c1\ud3c9\uade0\uac12 \uc815\ub9ac)<\/span><\/p>\n

                    \ud568\uc218 \\(f\\)\uac00 \\([a,\\, b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \ud568\uc218 \\(g \\geq 0\\)\uc774 \uc801\ubd84 \uac00\ub2a5\ud558\uba74, \ub2e4\uc74c \ub4f1\uc2dd\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac12 \\(c \\in [a,\\, b]\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.
                    \n\\[\\int_a^b f(x)g(x) dx = f(c)\\int_a^b g(x) dx.\\]<\/p>\n<\/div>\n

                    \n

                    \uc99d\uba85<\/p>\n

                    \\(f\\)\uac00 \uc5f0\uc18d\uc774\uace0 \\(g \\geq 0\\)\uc774 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790.
                    \n\\(f\\)\uac00 \uc5f0\uc18d\uc774\ubbc0\ub85c \\([a, b]\\)\uc5d0\uc11c \ucd5c\ub313\uac12 \\(M\\)\uacfc \ucd5c\uc19f\uac12 \\(m\\)\uc744 \uac00\uc9c4\ub2e4.
                    \n\\(g(x) \\geq 0\\)\uc774\ubbc0\ub85c \\(mg(x) \\leq f(x)g(x) \\leq Mg(x)\\)\uc774\ub2e4.
                    \n\uc774 \ubd80\ub4f1\uc2dd\uc758 \uac01 \ubcc0\uc744 \uc801\ubd84\ud558\uba74
                    \n\\[m\\int_a^b g(x) dx \\leq \\int_a^b f(x)g(x) dx \\leq M\\int_a^b g(x) dx\\]
                    \n\uc774\ub2e4. \ub9cc\uc57d \\(\\int_a^b g(x) dx = 0\\)\uc774\uba74, \\(\\int_a^b f(x)g(x) dx = 0\\)\uc774\uace0 \uc784\uc758\uc758 \\(c\\)\uc5d0 \ub300\ud574 \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.
                    \n\ub9cc\uc57d \\(\\int_a^b g(x) dx > 0\\)\uc774\uba74,
                    \n\\[m \\leq \\frac{\\int_a^b f(x)g(x) dx}{\\int_a^b g(x) dx} \\leq M\\]
                    \n\uc774\ubbc0\ub85c \uc0ac\uc787\uac12 \uc815\ub9ac\uc5d0 \uc758\ud574 \uc5b4\ub5a4 \\(c \\in [a, b]\\)\uc5d0 \ub300\ud574
                    \n\\[f(c) = \\frac{\\int_a^b f(x)g(x) dx}{\\int_a^b g(x) dx}\\]
                    \n\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

                    \n

                    \uc815\ub9ac 6.9. (\uc81c2\ud3c9\uade0\uac12 \uc815\ub9ac)<\/span><\/p>\n

                    \ud568\uc218 \\(f\\)\uac00 \ub2e8\uc870\ud568\uc218\uc774\uace0 \ubbf8\ubd84 \uac00\ub2a5\ud558\uba70 \ud568\uc218 \\(g\\)\uac00 \uc5f0\uc18d\uc774\uba74, \ub2e4\uc74c \ub4f1\uc2dd\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac12 \\(c \\in [a,\\, b]\\)\uac00 \uc874\uc7ac\ud55c\ub2e4.
                    \n\\[\\int_a^b f(x)g(x) dx = f(a)\\int_a^c g(x) dx + f(b)\\int_c^b g(x) dx.\\]<\/p>\n<\/div>\n

                    \n

                    \uc99d\uba85<\/p>\n

                    \uc77c\ubc18\uc131\uc744 \uc783\uc9c0 \uc54a\uace0 \\(f\\)\uac00 \ub2e8\uc870\uac10\uc18c\ud55c\ub2e4\uace0 \uac00\uc815\ud558\uc790.
                    \n\ud568\uc218 \\(G\\)\ub97c \\(G(x) = \\int_a^x g(t) dt\\)\ub77c\uace0 \uc815\uc758\ud558\uba74 \\(G(a) = 0\\)\uc774\uace0 \\(G'(x) = g(x)\\)\uc774\ub2e4.
                    \n\ubd80\ubd84\uc801\ubd84\ubc95\uc744 \uc0ac\uc6a9\ud558\uba74
                    \n\\[\\int_a^b f(x)g(x) dx = [f(x)G(x)]_a^b – \\int_a^b G(x)f'(x) dx\\]
                    \n\uc774\ub2e4.
                    \n\\(f\\)\uac00 \ub2e8\uc870\uac10\uc18c\uc774\ubbc0\ub85c \uc81c1\ud3c9\uade0\uac12 \uc815\ub9ac\uc5d0 \uc758\ud574 \uc801\ub2f9\ud55c \\(c \\in [a, b]\\)\uc5d0 \ub300\ud574
                    \n\\[\\int_a^b f(x)g(x) dx = f(a)G(c) + f(b)[G(b) – G(c)]\\]
                    \n\uc774\ub2e4.
                    \n\\(G(c) = \\int_a^c g(x) dx\\)\uc774\uace0 \\(G(b) – G(c) = \\int_c^b g(x) dx\\)\uc774\ubbc0\ub85c
                    \n\\[\\int_a^b f(x)g(x) dx = f(a)\\int_a^c g(x) dx + f(b)\\int_c^b g(x) dx\\]
                    \n\uc774\ub2e4.<\/span><\/p>\n<\/div>\n

                    \uc774\uc0c1\uc801\ubd84<\/h3>\n

                    \uc801\ubd84 \uad6c\uac04\uc774 \uc720\uacc4\uac00 \uc544\ub2c8\uac70\ub098, \ud568\uc218\uac00 \uc720\uacc4\uac00 \uc544\ub2d0 \ub54c\ub3c4 \uc801\ubd84\uc744 \uc815\uc758\ud560 \uc218 \uc788\ub2e4. \uc774\ub7ec\ud55c \uc801\ubd84\uc744 \uc774\uc0c1\uc801\ubd84<\/span>(improper integral)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

                    \uc720\uacc4\uac00 \uc544\ub2cc \uad6c\uac04\uc5d0\uc11c\uc758 \uc801\ubd84<\/h4>\n

                    \ud568\uc218 \\(f\\)\uac00 \uad6c\uac04 \\([a,\\, \\infty)\\)\uc758 \uc784\uc758\uc758 \ubd80\ubd84 \uc720\ud55c\uad6c\uac04\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uace0, \uadf9\ud55c
                    \n\\[\\int_a^{\\infty} f(x) dx = \\lim_{b \\to \\infty} \\int_a^b f(x) dx\\tag{6.6}\\]
                    \n\uac00 \uc874\uc7ac\ud558\uba74, “\\([a,\\,\\infty)\\)\uc5d0\uc11c \\(f\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834<\/span>(converge)\ud55c\ub2e4” \ub610\ub294 “\uc774\uc0c1\uc801\ubd84\uc774 \uc874\uc7ac\ud55c\ub2e4”\ub77c\uace0 \ub9d0\ud558\uace0, \uc774 \uadf9\ud55c\uac12\uc744 \\([a,\\,\\infty)\\)\uc5d0\uc11c \\(f\\)\uc758 \uc801\ubd84\uac12\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n

                    \uadf9\ud55c (6.6)\uc774 \uc218\ub834\ud558\uc9c0 \uc54a\uc73c\uba74, “\\([a,\\,\\infty)\\)\uc5d0\uc11c \\(f\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \ubc1c\uc0b0<\/span>(diverge)\ud55c\ub2e4”\ub77c\uace0 \ub9d0\ud55c\ub2e4.<\/p>\n

                    \uc774\uc0c1\uc801\ubd84 \\(\\int_{-\\infty}^b f(x) dx\\)\ub3c4 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n

                    \ub610\ud55c \uc784\uc758\uc758 \\(c\\)\uc5d0 \ub300\ud558\uc5ec
                    \n\\[\\int_{-\\infty}^{\\infty} f(x) dx = \\int_{-\\infty} ^c f(x) dx + \\int_{c}^{\\infty} f(x) dx\\tag{6.7}\\]
                    \n\ub85c \uc815\uc758\ud55c\ub2e4. \ub2e8, (6.7)\uc758 \uc6b0\ubcc0\uc758 \ub450 \uc774\uc0c1\uc801\ubd84\uc774 \ubaa8\ub450 \uc874\uc7ac\ud560 \ub54c\ub9cc \uc88c\ubcc0\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc874\uc7ac\ud558\ub294 \uac83\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \uc6b0\ubcc0\uc758 \ub450 \uc774\uc0c1\uc801\ubd84\uc774 \ubaa8\ub450 \uc874\uc7ac\ud558\ub294 \uacbd\uc6b0, \uc6b0\ubcc0\uc758 \ud569\uc740 \\(c\\)\uc758 \uac12\uc5d0 \uc0c1\uad00\uc5c6\uc774 \uc77c\uc815\ud558\ub2e4.<\/p>\n

                    \uc720\uacc4\uac00 \uc544\ub2cc \ud568\uc218\uc758 \uc801\ubd84<\/h4>\n

                    \ud568\uc218 \\(f\\)\uac00 \\((a,\\, b]\\)\uc758 \uc784\uc758\uc758 \ub2eb\ud78c \ubd80\ubd84\uad6c\uac04\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\uc9c0\ub9cc, \uc784\uc758\uc758 \\(\\delta > 0\\)\uc5d0 \ub300\ud558\uc5ec \\(f\\)\uac00 \\((a,\\,\\delta )\\)\uc5d0\uc11c \uc720\uacc4\uac00 \uc544\ub2d0 \ub54c, \\((a,\\,b]\\)\uc5d0\uc11c \\(f\\)\uc758 \uc801\ubd84\uc744 \ub2e4\uc74c\uacfc \uac19\uc740 \uadf9\ud55c\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.
                    \n\\[\\int_a^b f(x) dx = \\lim_{c \\to a^+} \\int_c^b f(x) dx .\\]
                    \n\uc989, \uc6b0\ubcc0\uc758 \uadf9\ud55c\uc774 \uc218\ub834\ud560 \ub54c \uc88c\ubcc0\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\ub294 \uac83\uc73c\ub85c \uc815\uc758\ud55c\ub2e4. \uc801\ubd84 \uad6c\uac04\uc758 \uc624\ub978\ucabd \ub05d\uc810 \uadfc\ubc29\uc5d0\uc11c \uc720\uacc4\uac00 \uc544\ub2cc \ud568\uc218\uc5d0 \ub300\ud574\uc11c\ub3c4 \uac19\uc740 \ubc29\ubc95\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n

                    \ub9cc\uc57d \ub450 \uad6c\uac04 \\([a,\\,c)\\)\uc640 \\((c,\\,b]\\) \uac01\uac01\uc5d0\uc11c \\(f\\)\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc218\ub834\ud558\uba74
                    \n\\[\\int_a^b f(x)dx = \\int_a^c f(x)dx + \\int_c^b f(x)dx\\]
                    \n\ub85c \uc815\uc758\ud55c\ub2e4. \uc989 \uc6b0\ubcc0\uc758 \ub450 \uc774\uc0c1\uc801\ubd84\uc774 \ubaa8\ub450 \uc874\uc7ac\ud560 \ub54c\ub9cc \uc88c\ubcc0\uc758 \uc774\uc0c1\uc801\ubd84\uc774 \uc874\uc7ac\ud558\ub294 \uac83\uc73c\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n

                    \uc790\uc8fc \ub4f1\uc7a5\ud558\ub294 \uc774\uc0c1\uc801\ubd84\uc740 \ub2e4\uc74c\uacfc \uac19\uc740 \uac83\ub4e4\uc774 \uc788\ub2e4.<\/p>\n