{"id":9497,"date":"2025-10-20T19:00:04","date_gmt":"2025-10-20T10:00:04","guid":{"rendered":"https:\/\/sasamath.com\/blog\/?page_id=9497"},"modified":"2026-09-27T16:25:11","modified_gmt":"2026-09-27T07:25:11","slug":"ch10-multiple-integral","status":"publish","type":"page","link":"https:\/\/sasamath.com\/blog\/invitation-to-mathematical-analysis\/ch10-multiple-integral\/","title":{"rendered":"\uc911\uc801\ubd84"},"content":{"rendered":"
<\/p>\n

<\/p>\n

\uc774 \uc7a5\uc5d0\uc11c\ub294 \ub9ac\ub9cc \ub2e4\uc911\uc801\ubd84\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \uc9c1\uc0ac\uac01\ud615 \uc601\uc5ed\uc5d0\uc11c\uc758 \uc801\ubd84\ubd80\ud130 \uc2dc\uc791\ud558\uc5ec \uc77c\ubc18 \uc601\uc5ed\uc73c\ub85c \ud655\uc7a5\ud558\uace0, \ud478\ube44\ub2c8 \uc815\ub9ac\uc640 \ubcc0\uc218\ubcc0\ud658 \uc815\ub9ac\ub97c \ud1b5\ud574 \uc801\ubd84\uc744 \uacc4\uc0b0\ud558\ub294 \ubc29\ubc95\uc744 \uc0b4\ud3b4\ubcf8\ub2e4. \ub9c8\uc9c0\ub9c9\uc5d0\ub294 \uc774\uc0c1\uc911\uc801\ubd84\uacfc \uac10\ub9c8\ud568\uc218, \ubca0\ud0c0\ud568\uc218\ub97c \uac04\ub2e8\ud788 \ub2e4\ub8ec\ub2e4.<\/p>\n

\ub2e4\uc911\uc801\ubd84\uc758 \uc815\uc758<\/h3>\n

\\(\\mathbb{R}^n\\)\uc5d0\uc11c \\(a_i<b_i\\)\uc774\uace0
\n\\[
\nR=[a_1,b_1]\\times\\cdots\\times[a_n,b_n]
\n\\]
\n\ud615\ud0dc\uc778 \uc9d1\ud569\uc744 \uc9c1\uc0ac\uac01\ud615 \uc9d1\ud569<\/span>(rectangle) \ub610\ub294 \uac04\ub2e8\ud788 \uc9c1\uc0ac\uac01\ud615\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \uc9c1\uc0ac\uac01\ud615\uc758 \ubd80\ud53c<\/span>(volume)\ub97c
\n\\[
\n|R|=\\prod_{i=1}^n(b_i-a_i)
\n\\]
\n\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n

\uc9c1\uc0ac\uac01\ud615 \\(R\\)\uc758 \ubd84\ud560<\/span>(partition) \\(P\\)\ub294 \uac01 \uc88c\ud45c\ucd95\uc5d0 \ub300\ud55c \ubd84\ud560\ub4e4\uc758 \uacf1\uc774\ub2e4. \uc989 \uac01 \uad6c\uac04 \\([a_i,b_i]\\)\ub97c
\n\\[
\na_i=x_{i,0}<x_{i,1}<\\cdots<x_{i,m_i}=b_i
\n\\]
\n\ub85c \ubd84\ud560\ud558\uba74 \\(R\\)\uc740 \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615
\n\\[
\nR_{j_1,\\ldots,j_n}
\n=[x_{1,j_1-1},x_{1,j_1}]\\times\\cdots\\times[x_{n,j_n-1},x_{n,j_n}]
\n\\]
\n\ub4e4\ub85c \ub098\ub25c\ub2e4. \uc11c\ub85c \ub2e4\ub978 \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615\uc758 \ub0b4\ubd80\ub294 \uc11c\ub85c\uc18c\uc774\uace0, \ubaa8\ub4e0 \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615\uc758 \ud569\uc9d1\ud569\uc740 \\(R\\)\uc774\ub2e4.<\/p>\n

\uc9c1\uc0ac\uac01\ud615 \\(R\\)\uc5d0\uc11c \uc720\uacc4\uc778 \ud568\uc218 \\(f\\colon R\\to\\mathbb{R}\\)\uc5d0 \ub300\ud558\uc5ec, \ubd84\ud560 \\(P\\)\uc758 \uac01 \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615 \\(S\\)\uc5d0\uc11c
\n\\[
\nm_S=\\inf\\{f(x)\\mid x\\in S\\},\\quad
\nM_S=\\sup\\{f(x)\\mid x\\in S\\}
\n\\]
\n\ub77c\uace0 \ud558\uc790. \ub9ac\ub9cc \uc0c1\ud569<\/span>\uacfc \ub9ac\ub9cc \ud558\ud569<\/span>\uc744
\n\\[
\nU(f,P)=\\sum_S M_S|S|,
\n\\quad
\nL(f,P)=\\sum_S m_S|S|
\n\\]
\n\ub85c \uc815\uc758\ud55c\ub2e4. \uc0c1\uc801\ubd84\uacfc \ud558\uc801\ubd84\uc740 \uac01\uac01
\n\\[
\n\\overline{\\int_R}f(x)\\,dx
\n=\\inf_P U(f,P),
\n\\quad
\n\\underline{\\int_R}f(x)\\,dx
\n=\\sup_P L(f,P)
\n\\]
\n\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n

\ud568\uc218 \\(f\\)\uac00 \\(R\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5<\/span>\ud558\ub2e4(Riemann integrable)\ub294 \uac83\uc740 \\(f\\)\uac00 \uc720\uacc4\uc774\uace0 \uc0c1\uc801\ubd84\uacfc \ud558\uc801\ubd84\uc774 \uac19\uc740 \uac83\uc774\ub2e4. \uc774 \uacf5\ud1b5\uac12\uc744
\n\\[
\n\\int_R f(x)\\,dx
\n\\]
\n\ub85c \ub098\ud0c0\ub0b4\uba70, \uac04\ub2e8\ud788 \\(\\int_R f\\)\ub77c\uace0 \uc4f0\uae30\ub3c4 \ud55c\ub2e4. \\(R\\subseteq\\mathbb{R}^2\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \\(\\iint_R f(x,y)\\,dA\\), \\(R\\subseteq\\mathbb{R}^3\\)\uc778 \uacbd\uc6b0\uc5d0\ub294 \\(\\iiint_R f(x,y,z)\\,dV\\)\ub77c\uace0 \uc4f4\ub2e4.<\/p>\n

\uc77c\ubcc0\uc218\uc758 \uacbd\uc6b0\uc640 \ub9c8\ucc2c\uac00\uc9c0\ub85c \ub2e4\uc74c \ud310\uc815\ubc95\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

\n

\uc815\ub9ac 10.1. (\ub2e4\uc911\uc801\ubd84\uc758 \ub9ac\ub9cc \ud310\uc815\ubc95)<\/span><\/p>\n

\uc9c1\uc0ac\uac01\ud615 \\(R\\)\uc5d0\uc11c \uc720\uacc4\uc778 \ud568\uc218 \\(f\\)\uac00 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud560 \ud544\uc694\ucda9\ubd84\uc870\uac74\uc740 \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec \\(R\\)\uc758 \ubd84\ud560 \\(P\\)\uac00 \uc874\uc7ac\ud558\uc5ec
\n\\[
\nU(f,P)-L(f,P)<\\varepsilon
\n\\]
\n\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \uac83\uc774\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/span>
\n\uc77c\ubcc0\uc218\uc758 \uc815\ub9ac 6.1\uc758 \ub9ac\ub9cc \ud310\uc815\ubc95<\/a>\uc758 \uc99d\uba85\uacfc \uac19\ub2e4. \\(f\\)\uac00 \uc801\ubd84 \uac00\ub2a5\ud558\uba74 \uc0c1\uc801\ubd84\uacfc \ud558\uc801\ubd84\uc758 \uacf5\ud1b5\uac12\uc744 \\(I\\)\ub77c\uace0 \ub450\uace0,
\n\\[
\nU(f,P_1)<I+\\frac{\\varepsilon}{2},
\n\\qquad
\nL(f,P_2)>I-\\frac{\\varepsilon}{2}
\n\\]
\n\uc778 \ub450 \ubd84\ud560\uc744 \ud0dd\ud55c\ub2e4. \ub450 \ubd84\ud560\uc758 \uacf5\ud1b5\uc138\ubd84 \\(P\\)\uc5d0 \ub300\ud558\uc5ec \uc0c1\ud569\uc740 \uc99d\uac00\ud558\uc9c0 \uc54a\uace0 \ud558\ud569\uc740 \uac10\uc18c\ud558\uc9c0 \uc54a\uc73c\ubbc0\ub85c
\n\\[
\nU(f,P)-L(f,P)<\\varepsilon
\n\\]
\n\uc774\ub2e4.<\/p>\n

\uc5ed\uc73c\ub85c \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec \uc774\ub7ec\ud55c \ubd84\ud560\uc774 \uc874\uc7ac\ud558\uba74
\n\\[
\n0\\le
\n\\overline{\\int_R}f-\\underline{\\int_R}f
\n\\le U(f,P)-L(f,P)<\\varepsilon
\n\\]
\n\uc774\ubbc0\ub85c \uc0c1\uc801\ubd84\uacfc \ud558\uc801\ubd84\uc774 \uac19\ub2e4.<\/span><\/p>\n<\/div>\n

\uc9c1\uc0ac\uac01\ud615\uc774 \uc544\ub2cc \uc720\uacc4\uc9d1\ud569\uc5d0\uc11c \uc801\ubd84\uc744 \uc815\uc758\ud558\uae30 \uc704\ud574 \ud2b9\uc131\ud568\uc218<\/span>(characteristic function)\ub97c \uc0ac\uc6a9\ud55c\ub2e4. \uc9d1\ud569 \\(E\\subseteq\\mathbb{R}^n\\)\uc758 \ud2b9\uc131\ud568\uc218\ub97c
\n\\[
\n\\chi_E(x)=
\n\\begin{cases}
\n1,&x\\in E,\\\\
\n0,&x\\notin E
\n\\end{cases}
\n\\]
\n\ub85c \uc815\uc758\ud55c\ub2e4.<\/p>\n

\uc720\uacc4\uc9d1\ud569 \\(E\\subseteq\\mathbb{R}^n\\)\uc640 \uc720\uacc4\ud568\uc218 \\(f\\colon E\\to\\mathbb{R}\\)\uac00 \uc8fc\uc5b4\uc84c\ub2e4\uace0 \ud558\uc790. \\(E\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc9c1\uc0ac\uac01\ud615 \\(R\\)\uc5d0\uc11c \\(f\\)\uc758 \uc601\uc5f0\uc7a5<\/span>(zero extension) \\(\\widetilde f\\colon R\\to\\mathbb{R}\\)\uc744
\n\\[
\n\\widetilde f(x)=
\n\\begin{cases}
\nf(x),&x\\in E,\\\\
\n0,&x\\in R\\setminus E
\n\\end{cases}
\n\\]
\n\ub85c \uc815\uc758\ud55c\ub2e4. \\(\\widetilde f\\)\uac00 \\(R\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud560 \ub54c \\(f\\)\uac00 \\(E\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uace0
\n\\[
\n\\int_E f(x)\\,dx=\\int_R\\widetilde f(x)\\,dx
\n\\]
\n\ub85c \uc815\uc758\ud55c\ub2e4. \uc774 \uc815\uc758\ub294 \\(E\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc9c1\uc0ac\uac01\ud615 \\(R\\)\uc758 \uc120\ud0dd\uacfc \ubb34\uad00\ud558\ub2e4. \uc2e4\uc81c\ub85c \ub354 \ud070 \uc9c1\uc0ac\uac01\ud615\uc73c\ub85c \uc62e\uaca8\ub3c4 \uc0c8\ub85c \ub354\ud574\uc9c0\ub294 \ubd80\ubd84\uc5d0\uc11c\ub294 \uc601\uc5f0\uc7a5\uc774 \\(0\\)\uc774\ubbc0\ub85c, \ub450 \uc9c1\uc0ac\uac01\ud615\uc758 \uacbd\uacc4\ub97c \ubd84\ud560\uc120\uc5d0 \ud3ec\ud568\uc2dc\ud0a4\uba74 \uc0c1\ud569\uacfc \ud558\ud569\uc5d0 \uc0c8\ub85c \ub354\ud574\uc9c0\ub294 \ud56d\uc774 \ubaa8\ub450 \\(0\\)\uc774\ub2e4.<\/p>\n

\uc801\ubd84 \uac00\ub2a5\uc131\uacfc \uc801\ubd84\uc758 \uc131\uc9c8<\/h3>\n

\uc720\uacc4\uc9d1\ud569 \\(E\\subseteq\\mathbb{R}^n\\)\uac00 \uc870\ub974\ub2e8 \uac00\uce21<\/span>(Jordan measurable)\uc774\ub77c\ub294 \uac83\uc740 \\(\\chi_E\\)\uac00 \uc801\ubd84 \uac00\ub2a5\ud55c \uac83\uc744 \ub73b\ud55c\ub2e4. \uc774\ub54c
\n\\[
\n|E|=\\int_R\\chi_E
\n\\]
\n\ub97c \\(E\\)\uc758 \uc870\ub974\ub2e8 \uce21\ub3c4<\/span>(Jordan measure), \ubd80\ud53c \ub610\ub294 \uccb4\uc801\uc774\ub77c\uace0 \ubd80\ub978\ub2e4. \ud2b9\ud788 \\(|E|=0\\)\uc774\uba74 \\(E\\)\uac00 \uc870\ub974\ub2e8 \uce21\ub3c4 0<\/span>\uc774\ub77c\uace0 \ud55c\ub2e4.<\/p>\n

\n

\uc815\ub9ac 10.2.<\/span><\/p>\n

\uc9d1\ud569 \\(E\\subseteq\\mathbb{R}^n\\)\uac00 \ucef4\ud329\ud2b8\uc774\uace0 \uc870\ub974\ub2e8 \uac00\uce21\uc774\uba70 \ud568\uc218 \\(f\\colon E\\to\\mathbb{R}\\)\uac00 \uc5f0\uc18d\uc774\uba74 \\(f\\)\ub294 \\(E\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n<\/div>\n

\n

\uc99d\uba85<\/span>
\n\\(E=\\varnothing\\)\uc774\uba74 \uc790\uba85\ud558\ub2e4. \uc774\uc81c \\(E\\ne\\varnothing\\)\uc774\ub77c\uace0 \ud558\uc790. \\(E\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc9c1\uc0ac\uac01\ud615 \\(R\\)\uc744 \ud0dd\ud558\uace0 \\(f\\)\uc758 \uc601\uc5f0\uc7a5\uc744 \\(\\widetilde f\\)\ub77c\uace0 \ud558\uc790. \\(E\\)\uac00 \ucef4\ud329\ud2b8\uc774\uace0 \\(f\\)\uac00 \uc5f0\uc18d\uc774\ubbc0\ub85c
\n\\[
\nM=\\max_{x\\in E}|f(x)|
\n\\]
\n\uac00 \uc874\uc7ac\ud55c\ub2e4. \\(M=0\\)\uc774\uba74 \uc790\uba85\ud558\ubbc0\ub85c \\(M>0\\)\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

\uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc744 \ud0dd\ud55c\ub2e4. \\(\\chi_E\\)\uac00 \uc801\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c \uc815\ub9ac 10.1\uc5d0 \uc758\ud574 \uc5b4\ub5a4 \ubd84\ud560 \\(P_0\\)\uac00 \uc874\uc7ac\ud558\uc5ec
\n\\[
\nU(\\chi_E,P_0)-L(\\chi_E,P_0)
\n<
\n\\frac{\\varepsilon}{4M}
\n\\]
\n\uc774\ub2e4. \ub610\ud55c \\(f\\)\ub294 \ucef4\ud329\ud2b8 \uc9d1\ud569 \\(E\\)\uc5d0\uc11c \uade0\ub4f1\uc5f0\uc18d\uc774\ubbc0\ub85c \uc5b4\ub5a4 \\(\\delta>0\\)\uac00 \uc874\uc7ac\ud558\uc5ec \\(x,y\\in E\\), \\(\\lVert x-y\\rVert<\\delta\\)\uc774\uba74
\n\\[
\n|f(x)-f(y)|
\n<
\n\\frac{\\varepsilon}{2|R|}
\n\\]
\n\uc774\ub2e4. \\(P_0\\)\ub97c \ub354 \uc138\ubd84\ud558\uc5ec \ubaa8\ub4e0 \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615\uc758 \uc9c0\ub984\uc774 \\(\\delta\\)\ubcf4\ub2e4 \uc791\uc740 \ubd84\ud560 \\(P\\)\ub97c \ub9cc\ub4e0\ub2e4.<\/p>\n

\ubd84\ud560 \\(P\\)\uc758 \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615 \\(S\\)\ub97c \uc0dd\uac01\ud558\uc790. \\(S\\subseteq E\\)\uc774\uba74 \\(S\\)\uc5d0\uc11c \\(\\widetilde f\\)\uc758 \uc9c4\ub3d9\uc740 \\(\\varepsilon\/(2|R|)\\)\ubcf4\ub2e4 \uc791\uace0, \\(S\\cap E=\\varnothing\\)\uc774\uba74 \uc9c4\ub3d9\uc740 \\(0\\)\uc774\ub2e4. \ub098\uba38\uc9c0 \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615, \uc989 \\(E\\)\uc640 \\(R\\setminus E\\)\ub97c \ubaa8\ub450 \ub9cc\ub098\ub294 \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615\uc5d0\uc11c\uc758 \uc9c4\ub3d9\uc740 \\(2M\\) \uc774\ud558\uc774\ub2e4. \uc774\ub7ec\ud55c \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615\ub4e4\uc758 \ubd80\ud53c\uc758 \ud569\uc740
\n\\[
\nU(\\chi_E,P)-L(\\chi_E,P)
\n\\]
\n\uc640 \uac19\ub2e4. \ub530\ub77c\uc11c
\n\\[
\n\\begin{aligned}
\nU(\\widetilde f,P)-L(\\widetilde f,P)
\n&\\le
\n\\frac{\\varepsilon}{2|R|}|R|
\n+2M\\bigl(U(\\chi_E,P)-L(\\chi_E,P)\\bigr)\\\\
\n&<
\n\\frac{\\varepsilon}{2}
\n+
\n\\frac{\\varepsilon}{2}
\n=
\n\\varepsilon.
\n\\end{aligned}
\n\\]
\n\uc815\ub9ac 10.1\uc5d0 \uc758\ud574 \\(\\widetilde f\\)\ub294 \\(R\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/span><\/p>\n<\/div>\n

\ub2e4\ubcc0\uc218 \ub9ac\ub9cc \uc801\ubd84\uc758 \uae30\ubcf8 \uc131\uc9c8\uc744 \uc815\ub9ac\ud558\uba74 \ub2e4\uc74c\uacfc \uac19\ub2e4.<\/p>\n

\n

\uc815\ub9ac 10.3. (\uc911\uc801\ubd84\uc758 \uae30\ubcf8 \uc131\uc9c8)<\/span><\/p>\n

\ud544\uc694\ud55c \ubaa8\ub4e0 \uc9d1\ud569\uc774 \uc720\uacc4\uc774\uace0 \uc870\ub974\ub2e8 \uac00\uce21\uc774\uba70 \uc544\ub798\uc5d0 \ub098\ud0c0\ub098\ub294 \ud568\uc218\ub4e4\uc774 \ud574\ub2f9 \uc9d1\ud569\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790.<\/p>\n

    \n
  1. \uc120\ud615\uc131:<\/strong> \uc2e4\uc218 \\(\\alpha,\\beta\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \n\\int_E(\\alpha f+\\beta g)
    \n=
    \n\\alpha\\int_E f+\\beta\\int_E g.
    \n\\]<\/li>\n
  2. \ub2e8\uc870\uc131:<\/strong> \\(f\\le g\\)\uc774\uba74
    \n\\[
    \n\\int_E f\\le\\int_E g.
    \n\\]<\/li>\n
  3. \uc808\ub313\uac12 \ubd80\ub4f1\uc2dd:<\/strong>
    \n\\[
    \n\\left|\\int_E f\\right|
    \n\\le
    \n\\int_E|f|.
    \n\\]<\/li>\n
  4. \uc601\uc5ed\uc758 \uac00\ubc95\uc131:<\/strong> \\(E=E_1\\cup E_2\\)\uc774\uace0 \\(|E_1\\cap E_2|=0\\)\uc774\uba74
    \n\\[
    \n\\int_E f
    \n=
    \n\\int_{E_1}f+\\int_{E_2}f.
    \n\\]<\/li>\n
  5. \ud3c9\ud589\uc774\ub3d9 \ubd88\ubcc0\uc131:<\/strong> \\(a\\in\\mathbb{R}^n\\)\uc774\uace0 \\(E+a=\\{x+a\\mid x\\in E\\}\\)\uc774\uba74
    \n\\[
    \n\\int_{E+a}f(x-a)\\,dx
    \n=
    \n\\int_E f(x)\\,dx.
    \n\\]<\/li>\n<\/ol>\n<\/div>\n
    \n

    \uc99d\uba85<\/span>
    \n\uc9c1\uc0ac\uac01\ud615\uc5d0\uc11c \uba3c\uc800 \ud569\uc5d0 \ub300\ud55c \uc120\ud615\uc131\uc744 \ubcf4\uc774\uc790. \ub450 \ud568\uc218\uc5d0 \ub300\ud574 \uac01\uac01 \ub9ac\ub9cc \ud310\uc815\ubc95\uc744 \ub9cc\uc871\uc2dc\ud0a4\ub294 \ubd84\ud560\uc758 \uacf5\ud1b5\uc138\ubd84 \\(P\\)\ub97c \ud0dd\ud55c\ub2e4. \uac01 \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615 \\(S\\)\uc5d0\uc11c
    \n\\[
    \n\\operatorname{osc}_S(f+g)
    \n\\le
    \n\\operatorname{osc}_S(f)+\\operatorname{osc}_S(g)
    \n\\]
    \n\uc774\ubbc0\ub85c \\(f+g\\)\ub3c4 \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \ub610\ud55c \uac19\uc740 \ubd84\ud560\uc5d0\uc11c
    \n\\[
    \nL(f,P)+L(g,P)
    \n\\le
    \nL(f+g,P)
    \n\\le
    \nU(f+g,P)
    \n\\le
    \nU(f,P)+U(g,P).
    \n\\]
    \n\uacf5\ud1b5\uc138\ubd84\uc744 \ud0dd\ud560 \ub54c \\(f\\)\uc640 \\(g\\)\uc758 \uc0c1\ud569\uacfc \ud558\ud569\uc744 \uac01\uac01\uc758 \uc801\ubd84\uac12\uc5d0 \uc784\uc758\ub85c \uac00\uae5d\uac8c \ub9cc\ub4e4 \uc218 \uc788\uc73c\ubbc0\ub85c
    \n\\[
    \n\\int_R(f+g)
    \n=
    \n\\int_Rf+\\int_Rg.
    \n\\]
    \n\uc0c1\uc218\ubc30\uc5d0 \ub300\ud574\uc11c\ub294 \\(\\alpha\\ge0\\)\uc774\uba74 \uc0c1\ud55c\uacfc \ud558\ud55c\uc774 \\(\\alpha\\)\ubc30 \ub418\uace0, \\(\\alpha<0\\)\uc774\uba74 \uc0c1\ud55c\uacfc \ud558\ud55c\uc758 \uc5ed\ud560\uc774 \ubc14\ub00c\ubbc0\ub85c
    \n\\[
    \n\\int_R\\alpha f
    \n=
    \n\\alpha\\int_Rf.
    \n\\]
    \n\ub530\ub77c\uc11c \uccab \ubc88\uc9f8 \uba85\uc81c\uac00 \uc131\ub9bd\ud55c\ub2e4. \uc77c\ubc18 \uc601\uc5ed\uc5d0\uc11c\ub294 \uc601\uc5f0\uc7a5\uc5d0 \uc774 \uacb0\uacfc\ub97c \uc801\uc6a9\ud558\uba74 \ub41c\ub2e4.<\/p>\n

    \ub2e8\uc870\uc131\uc740 \\(f\\le g\\)\uc774\uba74 \ubaa8\ub4e0 \ubd84\ud560\uc5d0\uc11c \\(L(f,P)\\le L(g,P)\\), \\(U(f,P)\\le U(g,P)\\)\uc778 \ub370\uc11c \ub530\ub978\ub2e4. \uc808\ub313\uac12 \ubd80\ub4f1\uc2dd\uc740
    \n\\[
    \n-|f|\\le f\\le |f|
    \n\\]
    \n\uc640 \uc120\ud615\uc131, \ub2e8\uc870\uc131\uc744 \uc801\uc6a9\ud558\uba74 \uc5bb\ub294\ub2e4.<\/p>\n

    \uc774\uc81c \\(A\\)\uac00 \uc870\ub974\ub2e8 \uce21\ub3c4 \\(0\\)\uc774\uace0 \\(h\\)\uac00 \\(A\\)\uc5d0\uc11c \uc720\uacc4\ub77c\uace0 \ud558\uc790. \\(|h|\\le M\\)\uc778 \\(M>0\\)\uc744 \ud0dd\ud55c\ub2e4. \\(\\chi_A\\)\uc758 \uc801\ubd84\uac12\uc774 \\(0\\)\uc774\ubbc0\ub85c \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \nU(\\chi_A,P)
    \n<
    \n\\frac{\\varepsilon}{2M}
    \n\\]
    \n\uc778 \ubd84\ud560 \\(P\\)\ub97c \ud0dd\ud560 \uc218 \uc788\ub2e4. \\(A\\)\ub97c \ub9cc\ub098\uc9c0 \uc54a\ub294 \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615\uc5d0\uc11c\ub294 \\(h\\)\uc758 \uc601\uc5f0\uc7a5\uc774 \\(0\\)\uc774\uace0, \\(A\\)\ub97c \ub9cc\ub098\ub294 \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615\uc5d0\uc11c\uc758 \uc9c4\ub3d9\uc740 \\(2M\\) \uc774\ud558\uc774\ubbc0\ub85c
    \n\\[
    \nU(\\widetilde h,P)-L(\\widetilde h,P)
    \n\\le
    \n2M U(\\chi_A,P)
    \n<
    \n\\varepsilon.
    \n\\]
    \n\ub530\ub77c\uc11c \\(h\\)\ub294 \\(A\\)\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \ub610\ud55c
    \n\\[
    \n-M\\chi_A
    \n\\le
    \n\\widetilde h
    \n\\le
    \nM\\chi_A
    \n\\]
    \n\uc774\ubbc0\ub85c \ub2e8\uc870\uc131\uc5d0 \uc758\ud574 \\(\\int_Ah=0\\)\uc774\ub2e4. \uc9d1\ud569 \\(B\\subseteq E\\)\uc5d0 \ub300\ud558\uc5ec \\(f|_B\\)\uc758 \uc601\uc5f0\uc7a5\uc744 \\(\\widetilde f_B\\)\ub77c\uace0 \uc4f0\uba74
    \n\\[
    \n\\widetilde f_E
    \n=
    \n\\widetilde f_{E_1}
    \n+
    \n\\widetilde f_{E_2}
    \n–
    \n\\widetilde f_{E_1\\cap E_2}.
    \n\\]
    \n\uad50\uc9d1\ud569\uc758 \uc801\ubd84\uc740 \\(0\\)\uc774\ubbc0\ub85c \uc120\ud615\uc131\uc744 \uc801\uc6a9\ud558\uba74 \uc601\uc5ed\uc758 \uac00\ubc95\uc131\uc744 \uc5bb\ub294\ub2e4.<\/p>\n

    \ub9c8\uc9c0\ub9c9\uc73c\ub85c \ubd84\ud560\uc744 \ubca1\ud130 \\(a\\)\ub9cc\ud07c \ud3c9\ud589\uc774\ub3d9\ud558\uba74 \uac01 \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615\uc758 \ubd80\ud53c\uc640 \uadf8 \uc704\uc5d0\uc11c\uc758 \uc0c1\ud55c\u00b7\ud558\ud55c\uc774 \uadf8\ub300\ub85c \ubcf4\uc874\ub41c\ub2e4. \ub530\ub77c\uc11c \uc0c1\ud569\uacfc \ud558\ud569\uc774 \uc77c\uce58\ud558\uace0 \ud3c9\ud589\uc774\ub3d9 \ubd88\ubcc0\uc131\uc744 \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

    \n

    \ubb38\uc81c 10.1.<\/span>
    \n\\(E\\)\uac00 \\(\\mathbb{R}^2\\)\uc5d0\uc11c \uc138 \uc810 \\((0,0)\\), \\((2,3)\\), \\((4,0)\\)\uc744 \uaf2d\uc9d3\uc810\uc73c\ub85c \ud558\ub294 \uc0bc\uac01\ud615 \uc601\uc5ed\uc77c \ub54c
    \n\\[
    \n\\iint_E(x-2y)\\,dA
    \n\\]
    \n\ub97c \uacc4\uc0b0\ud558\uc2dc\uc624. \ub4a4\uc5d0\uc11c \ubc30\uc6b0\ub294 \ud478\ube44\ub2c8 \uc815\ub9ac\ub97c \uc0ac\uc6a9\ud574\ub3c4 \uc88b\ub2e4.<\/p>\n<\/div>\n

    \n

    \ubb38\uc81c 10.2.<\/span>
    \n\\(E_1,E_2\\)\uc640 \\(E_1\\cup E_2\\), \\(E_1\\cap E_2\\)\uac00 \ubaa8\ub450 \uc870\ub974\ub2e8 \uac00\uce21\uc774\uace0 \\(f\\)\uac00 \uc774 \uc9d1\ud569\ub4e4\uc5d0\uc11c \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790. \ub2e4\uc74c \ud3ec\ud568\ubc30\uc81c \uacf5\uc2dd\uc744 \uc99d\uba85\ud558\uc2dc\uc624.
    \n\\[
    \n\\int_{E_1\\cup E_2}f
    \n+
    \n\\int_{E_1\\cap E_2}f
    \n=
    \n\\int_{E_1}f
    \n+
    \n\\int_{E_2}f.
    \n\\]<\/p>\n<\/div>\n

    \n

    \uc815\ub9ac 10.4. (\uc911\uc801\ubd84\uc758 \ud3c9\uade0\uac12 \uc815\ub9ac)<\/span><\/p>\n

    \\(E\\)\uac00 \uacf5\uc9d1\ud569\uc774 \uc544\ub2cc \uc5f0\uacb0\ub41c \ucef4\ud329\ud2b8 \uc870\ub974\ub2e8 \uac00\uce21\uc9d1\ud569\uc774\uace0 \\(f\\colon E\\to\\mathbb{R}\\)\uac00 \uc5f0\uc18d\uc774\uba74 \uc5b4\ub5a4 \\(c\\in E\\)\uac00 \uc874\uc7ac\ud558\uc5ec
    \n\\[
    \n\\int_E f
    \n=
    \nf(c)|E|
    \n\\]
    \n\uac00 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n<\/div>\n

    \n

    \uc99d\uba85<\/span>
    \n\\(|E|=0\\)\uc774\uba74 \uc815\ub9ac 10.3\uc5d0 \uc758\ud574 \\(\\int_E f=0\\)\uc774\ubbc0\ub85c \uc784\uc758\uc758 \\(c\\in E\\)\uc5d0 \ub300\ud558\uc5ec \ub4f1\uc2dd\uc774 \uc131\ub9bd\ud55c\ub2e4.<\/p>\n

    \uc774\uc81c \\(|E|>0\\)\uc774\ub77c\uace0 \ud558\uc790. \uc815\ub9ac 4.7\uc758 \ucd5c\ub300 \ucd5c\uc18c \uc815\ub9ac<\/a>\uc5d0 \uc758\ud574 \\(f\\)\ub294 \\(E\\)\uc5d0\uc11c \ucd5c\uc19f\uac12 \\(m\\)\uacfc \ucd5c\ub313\uac12 \\(M\\)\uc744 \uac00\uc9c4\ub2e4. \ub2e8\uc870\uc131\uc5d0 \uc758\ud574
    \n\\[
    \nm|E|
    \n\\le
    \n\\int_E f
    \n\\le
    \nM|E|.
    \n\\]
    \n\ub530\ub77c\uc11c
    \n\\[
    \nm
    \n\\le
    \n\\frac1{|E|}\\int_E f
    \n\\le
    \nM.
    \n\\]
    \n\uc5f0\uc18d\ud568\uc218\ub294 \uc5f0\uacb0\uc9d1\ud569\uc744 \uc5f0\uacb0\uc9d1\ud569\uc73c\ub85c \ubcf4\ub0b4\ubbc0\ub85c \\(f(E)\\)\ub294 \\(m\\)\uacfc \\(M\\)\uc744 \ud3ec\ud568\ud558\ub294 \uad6c\uac04\uc774\ub2e4. \ub530\ub77c\uc11c \uc5b4\ub5a4 \\(c\\in E\\)\uc5d0 \ub300\ud558\uc5ec
    \n\\[
    \nf(c)
    \n=
    \n\\frac1{|E|}\\int_E f
    \n\\]
    \n\uc774\uace0, \ubc14\ub77c\ub294 \ub4f1\uc2dd\uc744 \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

    \ubc18\ubcf5\uc801\ubd84\uacfc \ud478\ube44\ub2c8 \uc815\ub9ac<\/h3>\n

    \uc9c1\uc0ac\uac01\ud615 \\(R=[a,b]\\times[c,d]\\)\uc5d0\uc11c \uc911\uc801\ubd84\uc744 \uacc4\uc0b0\ud558\ub294 \uae30\ubcf8 \ub3c4\uad6c\ub294 \ud478\ube44\ub2c8 \uc815\ub9ac\uc774\ub2e4. \ub2e4\uc74c \ud615\ud0dc\ub294 \uc5f0\uc18d\ud568\uc218\ubfd0 \uc544\ub2c8\ub77c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud55c \ud568\uc218\uc5d0\ub3c4 \uc801\uc6a9\ud560 \uc218 \uc788\ub2e4.<\/p>\n

    \n

    \uc815\ub9ac 10.5. (\ud478\ube44\ub2c8 \uc815\ub9ac: \uc9c1\uc0ac\uac01\ud615\uc758 \uacbd\uc6b0)<\/span><\/p>\n

    \ud568\uc218 \\(f\\colon R\\to\\mathbb{R}\\)\uac00 \uc9c1\uc0ac\uac01\ud615 \\(R=[a,b]\\times[c,d]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4\uace0 \ud558\uc790.<\/p>\n

      \n
    1. \ubaa8\ub4e0 \\(x\\in[a,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(y\\mapsto f(x,y)\\)\uac00 \\([c,d]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uba74
      \n\\[
      \nF(x)=\\int_c^d f(x,y)\\,dy
      \n\\]
      \n\ub294 \\([a,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uace0
      \n\\[
      \n\\iint_R f(x,y)\\,dA
      \n=
      \n\\int_a^b
      \n\\left(
      \n\\int_c^d f(x,y)\\,dy
      \n\\right)dx.
      \n\\]<\/li>\n
    2. \ubaa8\ub4e0 \\(y\\in[c,d]\\)\uc5d0 \ub300\ud558\uc5ec \\(x\\mapsto f(x,y)\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\uba74
      \n\\[
      \n\\iint_R f(x,y)\\,dA
      \n=
      \n\\int_c^d
      \n\\left(
      \n\\int_a^b f(x,y)\\,dx
      \n\\right)dy.
      \n\\]<\/li>\n<\/ol>\n

      \ud2b9\ud788 \\(f\\)\uac00 \\(R\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba74 \ub450 \uc870\uac74\uc774 \ubaa8\ub450 \uc131\ub9bd\ud558\ubbc0\ub85c
      \n\\[
      \n\\iint_R f(x,y)\\,dA
      \n=
      \n\\int_a^b\\left(\\int_c^d f(x,y)\\,dy\\right)dx
      \n=
      \n\\int_c^d\\left(\\int_a^b f(x,y)\\,dx\\right)dy.
      \n\\tag{10.1}
      \n\\]
      \n\uc6b0\ubcc0\uc758 \ub450 \uc801\ubd84\uc744 \ubc18\ubcf5\uc801\ubd84<\/span>\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n

      \n

      \uc99d\uba85<\/span>
      \n\uccab \ubc88\uc9f8 \uba85\uc81c\ub97c \ubcf4\uc774\uba74 \ub450 \ubc88\uc9f8 \uba85\uc81c\ub294 \\(x\\)\uc640 \\(y\\)\uc758 \uc5ed\ud560\uc744 \ubc14\uafb8\uc5b4 \uc5bb\uc744 \uc218 \uc788\ub2e4. \\([a,b]\\)\uc758 \ubd84\ud560\uc744
      \n\\[
      \na=x_0<\\cdots<x_m=b,
      \n\\]
      \n\\([c,d]\\)\uc758 \ubd84\ud560\uc744
      \n\\[
      \nc=y_0<\\cdots<y_n=d
      \n\\]
      \n\ub77c\uace0 \ud558\uace0
      \n\\[
      \nI_i=[x_{i-1},x_i],
      \n\\qquad
      \nJ_j=[y_{j-1},y_j]
      \n\\]
      \n\ub77c\uace0 \ud558\uc790. \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615 \\(I_i\\times J_j\\)\uc5d0\uc11c \\(f\\)\uc758 \ud558\ud55c\uacfc \uc0c1\ud55c\uc744 \uac01\uac01 \\(m_{ij}\\), \\(M_{ij}\\)\ub77c\uace0 \ud558\uc790.<\/p>\n

      \uace0\uc815\ub41c \\(x\\in I_i\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[
      \nm_{ij}
      \n\\le
      \nf(x,y)
      \n\\le
      \nM_{ij}
      \n\\qquad
      \n(y\\in J_j)
      \n\\]
      \n\uc774\ubbc0\ub85c \uc77c\ubcc0\uc218 \uc801\ubd84\uc758 \ub2e8\uc870\uc131\uacfc \uad6c\uac04 \uac00\ubc95\uc131\uc5d0 \uc758\ud574
      \n\\[
      \n\\sum_{j=1}^n m_{ij}|J_j|
      \n\\le
      \nF(x)
      \n\\le
      \n\\sum_{j=1}^n M_{ij}|J_j|.
      \n\\]
      \n\ub530\ub77c\uc11c \\(I_i\\)\uc5d0\uc11c \\(F\\)\uc758 \ud558\ud55c\uacfc \uc0c1\ud55c\uc744 \uac01\uac01 \\(m_i^F\\), \\(M_i^F\\)\ub77c\uace0 \ud558\uba74
      \n\\[
      \n\\sum_jm_{ij}|J_j|
      \n\\le
      \nm_i^F
      \n\\le
      \nM_i^F
      \n\\le
      \n\\sum_jM_{ij}|J_j|.
      \n\\]
      \n\uc591\ubcc0\uc5d0 \\(|I_i|\\)\ub97c \uacf1\ud558\uc5ec \\(i\\)\uc5d0 \ub300\ud574 \ud569\ud558\uba74
      \n\\[
      \nL(f,P)
      \n\\le
      \nL(F,P_x)
      \n\\le
      \nU(F,P_x)
      \n\\le
      \nU(f,P),
      \n\\]
      \n\uadf8\ub9ac\uace0 \ud2b9\ud788
      \n\\[
      \nU(F,P_x)-L(F,P_x)
      \n\\le
      \nU(f,P)-L(f,P)
      \n\\]
      \n\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

      \\(f\\)\uac00 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ubbc0\ub85c \uc784\uc758\uc758 \\(\\varepsilon>0\\)\uc5d0 \ub300\ud558\uc5ec
      \n\\[
      \nU(f,P)-L(f,P)<\\varepsilon
      \n\\]
      \n\uc778 \uacf1\ubd84\ud560 \\(P=P_x\\times P_y\\)\ub97c \ud0dd\ud560 \uc218 \uc788\ub2e4. \uc704 \ubd80\ub4f1\uc2dd\uc5d0 \uc758\ud574 \\(F\\)\ub3c4 \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4. \ub610\ud55c \\(\\iint_R f\\)\uc640 \\(\\int_a^bF\\)\ub294 \ubaa8\ub450 \\([L(f,P),U(f,P)]\\) \uc548\uc5d0 \uc788\uc73c\ubbc0\ub85c \ub450 \uac12\uc758 \ucc28\uc774\ub294 \\(\\varepsilon\\)\ubcf4\ub2e4 \uc791\ub2e4. \\(\\varepsilon>0\\)\uc774 \uc784\uc758\uc774\ubbc0\ub85c
      \n\\[
      \n\\iint_R f(x,y)\\,dA
      \n=
      \n\\int_a^b
      \n\\left(
      \n\\int_c^d f(x,y)\\,dy
      \n\\right)dx.
      \n\\tag{10.2}
      \n\\]
      \n\uac19\uc740 \ub17c\ub9ac\ub85c
      \n\\[
      \n\\iint_R f(x,y)\\,dA
      \n=
      \n\\int_c^d
      \n\\left(
      \n\\int_a^b f(x,y)\\,dx
      \n\\right)dy.
      \n\\tag{10.3}
      \n\\]
      \n\ub97c \uc5bb\ub294\ub2e4.<\/span><\/p>\n<\/div>\n

      \ud478\ube44\ub2c8 \uc815\ub9ac\ub97c \ud568\uc218 \uadf8\ub798\ud504\ub85c \ub458\ub7ec\uc2f8\uc778 \uc601\uc5ed\uc5d0 \uc801\uc6a9\ud558\uba74 \ub2e4\uc74c \uacb0\uacfc\ub97c \uc5bb\ub294\ub2e4.<\/p>\n

      \n

      \uc815\ub9ac 10.6. (\ud478\ube44\ub2c8 \uc815\ub9ac: \uc9c1\uc0ac\uac01\ud615\uc774 \uc544\ub2cc \uc601\uc5ed)<\/span><\/p>\n

      \\(E\\subseteq\\mathbb{R}^2\\)\uac00 \ucef4\ud329\ud2b8 \uc870\ub974\ub2e8 \uac00\uce21\uc9d1\ud569\uc774\uace0 \\(f\\colon E\\to\\mathbb{R}\\)\uac00 \uc5f0\uc18d\uc774\ub77c\uace0 \ud558\uc790.<\/p>\n

        \n
      1. \n\\[
        \nE=
        \n\\{(x,y)\\mid
        \na\\le x\\le b,\\
        \n\\phi(x)\\le y\\le\\psi(x)\\}
        \n\\]
        \n\uc774\uace0 \\(\\phi,\\psi\\)\uac00 \\([a,b]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba70 \\(\\phi\\le\\psi\\)\uc774\uba74
        \n\\[
        \n\\iint_E f(x,y)\\,dA
        \n=
        \n\\int_a^b
        \n\\int_{\\phi(x)}^{\\psi(x)}
        \nf(x,y)\\,dy\\,dx.
        \n\\]<\/li>\n
      2. \n\\[
        \nE=
        \n\\{(x,y)\\mid
        \nc\\le y\\le d,\\
        \n\\alpha(y)\\le x\\le\\beta(y)\\}
        \n\\]
        \n\uc774\uace0 \\(\\alpha,\\beta\\)\uac00 \\([c,d]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uba70 \\(\\alpha\\le\\beta\\)\uc774\uba74
        \n\\[
        \n\\iint_E f(x,y)\\,dA
        \n=
        \n\\int_c^d
        \n\\int_{\\alpha(y)}^{\\beta(y)}
        \nf(x,y)\\,dx\\,dy.
        \n\\]<\/li>\n<\/ol>\n<\/div>\n
        \n

        \uc99d\uba85<\/span>
        \n\uccab \ubc88\uc9f8 \uacbd\uc6b0\ub97c \ubcf4\uc774\uc790. \\(E\\)\ub97c \ud3ec\ud568\ud558\ub294 \uc9c1\uc0ac\uac01\ud615 \\(R=[a,b]\\times[c,d]\\)\ub97c \ud0dd\ud558\uace0 \\(f\\)\uc758 \uc601\uc5f0\uc7a5\uc744 \\(\\widetilde f\\)\ub77c\uace0 \ud558\uc790. \uc815\ub9ac 10.2\uc5d0 \uc758\ud574 \\(\\widetilde f\\)\ub294 \\(R\\)\uc5d0\uc11c \ub9ac\ub9cc \uc801\ubd84 \uac00\ub2a5\ud558\ub2e4.<\/p>\n

        \uace0\uc815\ub41c \\(x\\in[a,b]\\)\uc5d0 \ub300\ud558\uc5ec \\(y\\mapsto\\widetilde f(x,y)\\)\ub294 \\([\\phi(x),\\psi(x)]\\)\uc5d0\uc11c \uc5f0\uc18d\uc774\uace0 \uadf8 \ubc14\uae65\uc5d0\uc11c\ub294 \\(0\\)\uc774\ub2e4. \ub530\ub77c\uc11c \ub9ce\uc544\uc57c \ub450 \uc810\uc5d0\uc11c\ub9cc \ubd88\uc5f0\uc18d\uc77c \uc218 \uc788\uc73c\ubbc0\ub85c \uc77c\ubcc0\uc218 \ub9ac\ub9cc \uc801\ubd84\uc774 \uc874\uc7ac\ud558\uace0
        \n\\[
        \n\\int_c^d\\widetilde f(x,y)\\,dy
        \n=
        \n\\int_{\\phi(x)}^{\\psi(x)}f(x,y)\\,dy.
        \n\\]
        \n\uc815\ub9ac 10.5\ub97c \uc801\uc6a9\ud558\uba74
        \n\\[
        \n\\int_E f
        \n=
        \n\\iint_R\\widetilde f(x,y)\\,dA
        \n=
        \n\\int_a^b
        \n\\left(
        \n\\int_{\\phi(x)}^{\\psi(x)}
        \nf(x,y)\\,dy
        \n\\right)dx.
        \n\\]
        \n\ub450 \ubc88\uc9f8 \uacbd\uc6b0\ub294 \\(x\\)\uc640 \\(y\\)\uc758 \uc5ed\ud560\uc744 \ubc14\uafb8\uba74 \ub41c\ub2e4.<\/span><\/p>\n<\/div>\n

        \uc77c\ubc18\uc801\uc73c\ub85c \uc601\uc5ed\uc774 \uc720\ud55c \uac1c\uc758 \uc774\ub7ec\ud55c \ud45c\uc900 \uc601\uc5ed\uc758 \ud569\uc9d1\ud569\uc73c\ub85c \ud45c\ud604\ub418\uace0 \uc11c\ub85c \ub2e4\ub978 \uc870\uac01\uc758 \uad50\uc9d1\ud569\uc774 \uc870\ub974\ub2e8 \uce21\ub3c4 \\(0\\)\uc774\uba74, \uc815\ub9ac 10.3\uc758 \uc601\uc5ed \uac00\ubc95\uc131\uc744 \uc0ac\uc6a9\ud558\uc5ec \uac01 \uc870\uac01\uc758 \ubc18\ubcf5\uc801\ubd84\uc744 \ud569\ud558\uba74 \ub41c\ub2e4.<\/p>\n

        \n

        \ubcf4\uae30 10.7.<\/span><\/p>\n

        \ud568\uc218 \\(f\\)\uac00 \uc6d0\ud310
        \n\\[
        \nD=\\{(x,y)\\mid x^2+y^2\\le r^2\\}
        \n\\]
        \n\uc5d0\uc11c \uc5f0\uc18d\uc774\ub77c\uace0 \ud558\uc790. \\(D\\)\uc5d0\uc11c\ub294 \ub450 \uc801\ubd84 \uc21c\uc11c\ub97c \ubaa8\ub450 \uc0ac\uc6a9\ud560 \uc218 \uc788\ub2e4.
        \n\\[
        \n\\iint_Df(x,y)\\,dA
        \n=
        \n\\int_{-r}^r
        \n\\int_{-\\sqrt{r^2-x^2}}^{\\sqrt{r^2-x^2}}
        \nf(x,y)\\,dy\\,dx,
        \n\\]
        \n\\[
        \n\\iint_Df(x,y)\\,dA
        \n=
        \n\\int_{-r}^r
        \n\\int_{-\\sqrt{r^2-y^2}}^{\\sqrt{r^2-y^2}}
        \nf(x,y)\\,dx\\,dy.
        \n\\]<\/p>\n<\/div>\n

        \uace0\ucc28\uc6d0\uc5d0\uc11c\ub3c4 \ud478\ube44\ub2c8 \uc815\ub9ac\ub97c \ubc18\ubcf5\ud558\uc5ec \uc801\uc6a9\ud560 \uc218 \uc788\ub2e4. \ud2b9\ud788 \uc5f0\uc18d\ud568\uc218\uc5d0 \ub300\ud574\uc11c\ub294 \uc9c1\uc0ac\uac01\ud615 \uc601\uc5ed\uc758 \uc911\uc801\ubd84\uc744 \uc88c\ud45c\ubcc4 \ubc18\ubcf5\uc801\ubd84\uc73c\ub85c \uacc4\uc0b0\ud560 \uc218 \uc788\ub2e4.<\/p>\n

        \n

        \ubcf4\uae30 10.8.<\/span><\/p>\n

        \uc0ac\uba74\uccb4
        \n\\[
        \nT=\\{(x,y,z)\\mid x,y,z\\ge0,\\ x+y+z\\le1\\}
        \n\\]
        \n\uc758 \ubd80\ud53c\ub294
        \n\\[
        \n\\begin{aligned}
        \n|T|
        \n&=
        \n\\int_0^1
        \n\\int_0^{1-x}
        \n\\int_0^{1-x-y}
        \ndz\\,dy\\,dx\\\\
        \n&=
        \n\\int_0^1
        \n\\int_0^{1-x}
        \n(1-x-y)\\,dy\\,dx\\\\
        \n&=
        \n\\int_0^1
        \n\\frac{(1-x)^2}{2}\\,dx
        \n=
        \n\\frac16.
        \n\\end{aligned}
        \n\\]<\/p>\n<\/div>\n

        \n

        \ubb38\uc81c 10.3.<\/span>
        \n\ud568\uc218 \\(f\\)\uc640 \uc9d1\ud569 \\(E\\)\uac00 \ub2e4\uc74c\uacfc \uac19\uc774 \uc8fc\uc5b4\uc84c\uc744 \ub54c \\(E\\) \uc704\uc5d0\uc11c \\(f\\)\uc758 \uc801\ubd84\uc744 \uacc4\uc0b0\ud558\uc2dc\uc624.<\/p>\n

          \n
        1. \\(f(x,y)=\\dfrac1{1+x^2}\\), \\(E\\)\ub294 \ub450 \uc9c1\uc120 \\(x=1\\), \\(y=0\\)\uacfc \uace1\uc120 \\(y=x^3\\)\uc73c\ub85c \ub458\ub7ec\uc2f8\uc778 \uc601\uc5ed.<\/li>\n
        2. \\(f(x,y)=x+y\\), \\(E\\)\ub294 \uc138 \uc810 \\((0,0)\\), \\((0,1)\\), \\((2,0)\\)\uc744 \uaf2d\uc9d3\uc810\uc73c\ub85c \ud558\ub294 \uc0bc\uac01\ud615 \uc601\uc5ed.<\/li>\n
        3. \\(f(x,y,z)=x\\), \\(E\\)\ub294 \\(0\\le z\\le1-x^2\\), \\(0\\le y\\le x^2+z^2\\)\ub97c \ub9cc\uc871\uc2dc\ud0a4\ub294 \uc810\uc758 \uc9d1\ud569.<\/li>\n<\/ol>\n<\/div>\n
          \n

          \ubb38\uc81c 10.4.<\/span>
          \n\\(D\\)\uac00 \\(y=x^2\\)\uacfc \\(y=2x\\)\ub85c \ub458\ub7ec\uc2f8\uc778 \uc601\uc5ed\uc77c \ub54c
          \n\\[
          \n\\iint_Dxy\\,dA
          \n\\]
          \n\ub97c \uacc4\uc0b0\ud558\uc2dc\uc624.<\/p>\n<\/div>\n

          \ubcc0\uc218\ubcc0\ud658<\/h3>\n

          \uc77c\ubcc0\uc218 \uc801\ubd84\uc758 \uce58\ud658\uc801\ubd84\uc5d0 \ud574\ub2f9\ud558\ub294 \uac83\uc774 \ub2e4\ubcc0\uc218 \uc801\ubd84\uc758 \ubcc0\uc218\ubcc0\ud658 \uc815\ub9ac\uc774\ub2e4. 9\uc7a5<\/a>\uc5d0\uc11c \uc815\uc758\ud55c \ubbf8\ubd84\ub3d9\ud615\uc0ac\uc0c1\uc744 \uc0ac\uc6a9\ud558\uc5ec \uc815\ub9ac\ub97c \uc9c4\uc220\ud560 \uc218 \uc788\ub2e4.<\/p>\n

          \n

          \uc815\ub9ac 10.9. (\ubcc0\uc218\ubcc0\ud658 \uc815\ub9ac)<\/span><\/p>\n

          \\(U,V\\subseteq\\mathbb{R}^n\\)\uac00 \uc5f4\ub9b0\uc9d1\ud569\uc774\uace0 \\(\\phi\\colon U\\to V\\)\uac00 \ubbf8\ubd84\ub3d9\ud615\uc0ac\uc0c1\uc774\ub77c\uace0 \ud558\uc790. \\(K\\subseteq U\\)\uac00 \ucef4\ud329\ud2b8 \uc870\ub974\ub2e8 \uac00\uce21\uc9d1\ud569\uc774\uace0 \\(f\\colon\\phi(K)\\to\\mathbb{R}\\)\uac00 \uc5f0\uc18d\uc774\uba74 \\(\\phi(K)\\)\ub3c4 \uc870\ub974\ub2e8 \uac00\uce21\uc774\uace0
          \n\\[
          \n\\int_{\\phi(K)}f(y)\\,dy
          \n=
          \n\\int_K
          \nf(\\phi(x))\\,|\\det D\\phi(x)|\\,dx.
          \n\\tag{10.4}
          \n\\]
          \n\uc5ec\uae30\uc11c \\(\\det D\\phi(x)\\)\ub97c \uc57c\ucf54\ube44 \ud589\ub82c\uc2dd<\/span>(Jacobian determinant)\uc774\ub77c\uace0 \ubd80\ub978\ub2e4.<\/p>\n<\/div>\n

          \ubcc0\uc218\ubcc0\ud658 \uc815\ub9ac\uc758 \uc99d\uba85\uc5d0\uc11c \ud575\uc2ec\uc774 \ub418\ub294 \ubd80\ud53c \ubcc0\ud654\ub294 \ub2e4\uc74c \uc138 \uc2dd\uc73c\ub85c \uc694\uc57d\ud560 \uc218 \uc788\ub2e4. \uac00\uc5ed\uc120\ud615\ubcc0\ud658 \\(A\\)\uc640 \uc791\uc740 \uc9c1\uc0ac\uac01\ud615 \\(Q\\)\uc5d0 \ub300\ud558\uc5ec
          \n\\[
          \n|A(Q)|
          \n=
          \n|\\det A|\\,|Q|.
          \n\\tag{10.5}
          \n\\]
          \n\ub610\ud55c \\(\\phi\\)\uac00 \\(C^1\\)\uc774\uba74 \ucef4\ud329\ud2b8 \uc9d1\ud569 \uc704\uc5d0\uc11c \uade0\uc77c\ud558\uac8c
          \n\\[
          \n\\phi(x+h)
          \n=
          \n\\phi(x)+D\\phi(x)h+o(\\lVert h\\rVert)
          \n\\qquad
          \n(h\\to0)
          \n\\tag{10.6}
          \n\\]
          \n\uc774\uace0, \uc774 \uc120\ud615\uadfc\uc0ac\ub97c \uc870\ub974\ub2e8 \uce21\ub3c4\uc5d0 \ub300\ud574 \uc815\ub7c9\ud654\ud558\uba74
          \n\\[
          \n\\frac{|\\phi(Q)|}{|Q|}
          \n=
          \n|\\det D\\phi(x)|+o(1)
          \n\\quad
          \n(Q\\text{\uac00 }x\\text{\ub97c \uc911\uc2ec\uc73c\ub85c \ud558\ub294 \uc815\uc721\uba74\uccb4\ub85c\uc11c }x\\text{\uc5d0 \uc218\ucd95\ud560 \ub54c})
          \n\\tag{10.7}
          \n\\]
          \n\ub77c\ub294 \uad6d\uc18c \ubd80\ud53c \uc65c\uace1 \ucd94\uc815\uc744 \uc5bb\ub294\ub2e4. \ub9c8\uc9c0\ub9c9 \uc2dd\uc758 \uc5c4\ubc00\ud55c \uc99d\uba85\uc774 \ubcc0\uc218\ubcc0\ud658 \uc815\ub9ac\uc758 \uae30\uc220\uc801\uc778 \ud575\uc2ec\uc774\ub2e4.<\/p>\n

          \ubcc0\uc218\ubcc0\ud658 \uc815\ub9ac\uc758 \uc5c4\ubc00\ud55c \uc99d\uba85\uc5d0\ub294 \\(C^1\\) \uc0ac\uc0c1\uc774 \uc791\uc740 \uc9c1\uc0ac\uac01\ud615\uc758 \ubd80\ud53c\ub97c \\(|\\det D\\phi|\\)\uc758 \ube44\uc728\ub85c \ubcc0\ud654\uc2dc\ud0a8\ub2e4\ub294 \uad6d\uc18c \ubd80\ud53c \uc65c\uace1 \ucd94\uc815\uacfc, \uadf8 \ucd94\uc815\uc744 \uc870\ub974\ub2e8 \ubd84\ud560 \uc804\uccb4\uc5d0 \uade0\uc77c\ud558\uac8c \uc801\uc6a9\ud558\ub294 \ub17c\uc99d\uc774 \ud544\uc694\ud558\ub2e4. \uc774 \uc99d\uba85\uc740 \uc774 \ucc45\uc5d0\uc11c \uc9c0\uae08\uae4c\uc9c0 \uc900\ube44\ud55c \ub3c4\uad6c\ubcf4\ub2e4 \uae30\uc220\uc801\uc774\ubbc0\ub85c \uc5ec\uae30\uc11c\ub294 \uc815\ub9ac 10.9\ub97c \uc99d\uba85 \uc5c6\uc774 \uc0ac\uc6a9\ud55c\ub2e4. \ub2e4\ub9cc \uc544\ud540 \ubcc0\ud658 \\(\\phi(x)=Ax+b\\)\uc758 \uacbd\uc6b0\uc5d0\ub294 \uc120\ud615\ub300\uc218\ud559\uc5d0\uc11c \uc54c\ub824\uc9c4 \ubd80\ud53c \uacf5\uc2dd
          \n\\[
          \n|A(R)|
          \n=
          \n|\\det A|\\,|R|
          \n\\]
          \n\uc744 \uac01 \ubd80\ubd84\uc9c1\uc0ac\uac01\ud615\uc5d0 \uc801\uc6a9\ud558\uba74 \uc704 \uc801\ubd84 \uacf5\uc2dd\uc774 \uc9c1\uc811 \ub530\ub978\ub2e4.<\/p>\n

          \ud45c\uc900 \uc88c\ud45c\ubcc0\ud658\uc740 \ub9e4\uac1c\ubcc0\uc218 \uc601\uc5ed\uc758 \uacbd\uacc4\ub098 \ucd95\uc5d0\uc11c \uc77c\ub300\uc77c\uc131\uc774 \uae68\uc9c8 \uc218 \uc788\ub2e4. \uc774\ub7ec\ud55c \uc608\uc678\uc9d1\ud569\uc740 \uc870\ub974\ub2e8 \uce21\ub3c4 \\(0\\)\uc774\uace0, \ub0b4\ubd80\ub97c \uc720\ud55c \uac1c\uc758 \uc870\uac01\uc73c\ub85c \ub098\ub204\uc5b4 \uc815\ub9ac 10.9\ub97c \uc801\uc6a9\ud55c \ub4a4 \uc601\uc5ed\uc758 \uac00\ubc95\uc131\uc744 \uc0ac\uc6a9\ud558\uba74 \uac19\uc740 \uacf5\uc2dd\uc744 \uc5bb\ub294\ub2e4. \uc774 \uc870\uac01\ubcc4 \ud655\uc7a5\ub3c4 \uc5ec\uae30\uc11c\ub294 \uc99d\uba85 \uc5c6\uc774 \uc0ac\uc6a9\ud55c\ub2e4. \ub2e4\uc74c \uc88c\ud45c\ubcc0\ud658\uc744 \uc790\uc8fc \uc0ac\uc6a9\ud55c\ub2e4.<\/p>\n