1. 二重积分

多变量函数的重积分

张瑞
中国科学技术大学数学科学学院

二重积分

\begin{tikzpicture} %draw the axes \draw[->] (0,0,0) -- (3,0,0) node[anchor=west]{$y$}; \draw[->] (0,0,0) -- (0,2,0) node[anchor=west]{$z$}; \draw[->] (0,0,0) -- (0,0,3) node[anchor=west]{$x$}; % 画曲面和定义域 \draw[dashed] (0.2, -0.4)--(1.4, -0.85)--(2.3,-0.25); \draw[dashed] (0.2, -0.4) .. controls (1,0) .. (2.3,-0.25); \draw[thick] (0.2, 1.4).. controls(0.75, 1.35) ..(1.4, 0.75); \draw[thick] (2.3, 1.25).. controls(1.7, 1.2)..(1.4, 0.75); \draw[thick] (0.2, 1.4) .. controls (1,2) .. (2.3,1.25); \draw[dashed] (0.2,1.4)--(0.2,-0.4); \draw[dashed] (2.3, 1.25)--(2.3, -0.25); \draw[dashed] (1.4, 0.75)--(1.4, -0.85); % 画小单元 \coordinate (a1) at (0.7, -0.4); \coordinate (a2) at (0.9, -0.6); \coordinate (a3) at (1.2, -0.35); \coordinate (c1) at (1.0, -0.3); \draw[dotted, red] (a1)--(a2)--(a3).. controls (c1) .. (a1); \coordinate (b1) at (0.7, 1.4); \coordinate (b2) at (0.9, 1.3); \coordinate (b3) at (1.2, 1.35); \coordinate (d1) at (0.8, 1.4); \coordinate (d2) at (1.1, 1.37); \coordinate (d3) at (1.0, 1.6); \filldraw[thick, red, draw=black] (b1).. controls (d1) ..(b2).. controls (d2) ..(b3).. controls (d3) .. (b1); \draw[dashed] (a1)--(b1); \draw[dashed] (a2)--(b2); \draw[dashed] (a3)--(b3); \end{tikzpicture}

类比与一维积分的Riemann和，来看看二维积分的Riemann和。

平面区域的面积

平面区域上的点集$D$的面积为一个非负实数，记为$\sigma(D)$，满足：

  1. 约定边长为1的正方形的面积为1。这样，边长为$l$的正方形的面积就是$l^2$。
  2. 如果$D$由互不重叠的有限个集合$D_1$, $\cdots$, $D_k$组成，则$D$的面积满足可加性
    \[\sigma(D)=\sigma(D_1)+\sigma(D_2)+\cdots+\sigma(D_k) \]
\begin{tikzpicture}[scale=1.3] % 网格 \draw[black!20, step=.25cm] (-0.4,-0.1) grid (2.7,2.4); %\node at (0,0) {$O$}; %\filldraw[red] (0,0)--(0,0.25)--(0.25, 0.25) --cycle; % 覆盖区域的网格 \filldraw[blue!50, opacity=0.5] (0.5,0.25)--(2.25,0.25)--(2.25, 2) -- (1,2) -- (1, 1.75)--(0.75, 1.75) -- (0.75, 1.5) -- (0.25, 1.5) --(0.25, 1.25) -- (0,1.25) -- (0,0.75) -- (0.25, 0.75) -- (0.25, 0.5) -- (0.5, 0.5)--cycle; \filldraw[blue!80, opacity=0.5] (0.5,1.25)--(0.5,0.75)--(0.75, 0.75) -- (0.75, 0.5) -- (2.0,0.5)--(2.0,1.75) -- (1.5, 1.75) -- (1.5,1.5) -- (0.75, 1.5) -- (0.75, 1.25) --cycle; % 区域边界 \draw[thick] plot[smooth cycle] coordinates {(0.2,1.1) (0.71, 0.42) (2.05, 0.4) (2.02,1.8) (1.6,1.9)}; \end{tikzpicture}

设$D$是有界集，则有$D\subset[a,b]\times[a,b]$。作分割

\[T: a=x_0<x_1<\cdots<x_n=b \]

这样，可以得到$n^2$个小正方形。

记完全包含在$D$内的小正方形的面积和为$\sigma_T^-(D)$，那些与$D$有交集的小正方形的面积和为$\sigma_T^+(D)$

  • 如果$D$有面积，则面积$\sigma(D)$介于$\sigma_T^-(D)$和$\sigma_T^+(D)$之间。
  • 不难看出，随着分割越来越细，$\sigma_T^-(D)$单调递增，$\sigma_T^+(D)$单调递减。因此， $|T|\to0$时，它们都有极限。
  • 如果极限相等，自然定义这个极限值为面积$\sigma(D)$。

定义 1.
设$D$是有界集，取$[a,b]\times[a,b]$使得$D\subset[a,b]\times[a,b]$。若对于任意分割

\[T: a=x_0<x_1<\cdots<x_n=b \]

有

\[\lim_{|T|\to0}\sigma_T^-(D) =\lim_{|T|\to0}\sigma_T^+(D) \]

则称点集$D$是Jordan可测的，否则称为不可测的。这个极限值称为$D$的测度或面积。 极限为零时，称$D$为零测集。

  • 矩形是可测的，它的测度是矩形的长和宽的乘积。 因此，在做分割时也可以使用矩形分割。
  • 记$D$的边界为$\partial D$，则包含边界点的小正方形面积满足
    \[0\leq \sigma_T^+(D)-\sigma_T^-(D)\leq\sigma_T^+(D) \]
    因此，当$\partial D$为零测集时，$D$是可测的。进一步有

    定理 1.
    $D$是Jordan可测的充分必要条件是它的边界$\partial D$是零测集。

例 1. (面积为0的点集) 闭区间上连续函数

\[y=f(x), x\in[a,b] \quad\mbox{or}\quad x=g(y), y\in[c,d] \]

给出的平面曲线段的面积为零。

例 2. (不可测度的点集) $[0,1]\times[0,1]$内所有有理点所构成的集合$D$是Jordan意义下不可测的。

解. 可以得到$\partial D=[0,1]\times[0,1]$，从而$\partial D$的面积不为0。

或者，依定义，对任意分割$T$，有

\[\sigma_T^-(D)=0, \quad \sigma_T^+(D)=1 \]
  • 如果一个有界点集的边界可以逐段表示成闭区间上连续函数给出的曲线段，则该点集是可测的。
  • 利用积分定义的曲边梯形的面积与Jordan测度是一致的。 Darboux上和是$\sigma_T^+(D)$，Darboux下和是$\sigma_T^-(D)$

二重积分的基本概念与性质

今后，假定平面区域$D$是闭的有界区域，$D$以及它的分割都是可测的。

定义 2.
所谓$D$的“分割”$T$，指把$D$分成$n$个互不重叠的可测区域$D_i$, $i=1,2,\cdots,n$。 记$\sigma(D_i)$表示分割小区域的面积，并记

\[|T|=\max_{1\leq i\leq n} \mbox{diam}( D_i) \]

这里$\mbox{diam}( D_i)$表示$D_i$的直径。称

\[S(T)=\sum_{i=1}^n f(\xi_i, \eta_i)\sigma(D_i) \]

为$f(x,y)$在$D$上的一个Riemann和。

\begin{tikzpicture} %draw the axes \draw[->] (0,0,0) -- (3,0,0) node[anchor=west]{$y$}; \draw[->] (0,0,0) -- (0,2,0) node[anchor=west]{$z$}; \draw[->] (0,0,0) -- (0,0,3) node[anchor=west]{$x$}; % 画曲面和定义域 \draw[dashed] (0.2, -0.4)--(1.4, -0.85)--(2.3,-0.25); \draw[dashed] (0.2, -0.4) .. controls (1,0) .. (2.3,-0.25); \draw[thick] (0.2, 1.4).. controls(0.75, 1.35) ..(1.4, 0.75); \draw[thick] (2.3, 1.25).. controls(1.7, 1.2)..(1.4, 0.75); \draw[thick] (0.2, 1.4) .. controls (1,2) .. (2.3,1.25); \draw[dashed] (0.2,1.4)--(0.2,-0.4); \draw[dashed] (2.3, 1.25)--(2.3, -0.25); \draw[dashed] (1.4, 0.75)--(1.4, -0.85); % 画小单元 \coordinate (a1) at (0.7, -0.4); \coordinate (a2) at (0.9, -0.6); \coordinate (a3) at (1.2, -0.35); \coordinate (c1) at (1.0, -0.3); \draw[dotted, red] (a1)--(a2)--(a3).. controls (c1) .. (a1); \coordinate (b1) at (0.7, 1.4); \coordinate (b2) at (0.9, 1.3); \coordinate (b3) at (1.2, 1.35); \coordinate (d1) at (0.8, 1.4); \coordinate (d2) at (1.1, 1.37); \coordinate (d3) at (1.0, 1.6); \filldraw[thick, red, draw=black] (b1).. controls (d1) ..(b2).. controls (d2) ..(b3).. controls (d3) .. (b1); \draw[dashed] (a1)--(b1); \draw[dashed] (a2)--(b2); \draw[dashed] (a3)--(b3); \end{tikzpicture}

定义 3.
若存在数$A$满足： $\forall\epsilon>0$，存在$\delta >0$，只要$|T|<\delta$， 不论$(\xi_i,\eta_i)$在$D_i$中如何选择，都有

\[\left|\sum_{i=1}^n f(\xi_i, \eta_i)\sigma(D_i)-A\right|<\epsilon \]

则称函数$f(x,y)$在$D$上可积，记为

\[A=\iint_D f(x,y)d\sigma \quad\mbox{or}\quad \int_D f \]

几何上看： $S(T)$是以$D$上的函数$f(x,y)$为顶的曲顶柱体的体积的近似，它与一元定积分是一样的。

物理上看： 把$f(x,y)$看成是面积为$D$的一块薄板的密度函数，则二重积分就是薄板的质量。

定理 2.
$D$上可积函数必有界

证明. 与单变量情形完全一样

设$f(x,y)$在$D$上有界：$m\leq f(x,y)\leq M$，记$f(x,y)$在小区域$D_i$的上、下确界为

\[m_i=\inf\{f(x,y)|(x,y)\in D_i\}, M_i=\sup\{f(x,y), (x,y)\in D_i\} \]

定义函数$f(x,y)$在$D$上的Darboux上和, Darboux下和

\[\bar S(T)=\sum_{i=1}^n M_i\sigma(D_i),\,\, \underline S(T)=\sum_{i=1}^n m_i\sigma(D_i) \]

振幅的和为

\[\omega(T)=\sum_{i=1}^n (M_i-m_i)\sigma(D_i)= \bar S(T)-\underline S(T) \]

类似一元定积分的分析过程，有

  1. Darboux上和有下界，Darboux下和有上界
    \[\bar S(T)\geq m\sigma(D), \quad \underline S(T)\leq M \sigma(D) \]
  2. $\bar S(T)\geq S(T)\geq\underline S(T)$
  3. $\displaystyle \inf_T\{\bar S(T)\}\geq \sup_T\{\bar S(T)\}$

定理 3.
$f(x,y)$在$D$上有界，则$f(x,y)$在$D$上可积的充分必要条件是Darboux上和的下确界与Darboux下和的上确界相等，或者$\displaystyle \inf_T\{\omega(T)\}=0$

定理 4.
$f(x,y)$是$D$上的有界函数。

  1. 若$f(x,y)$的不连续点分布在$D$中的可测的且测度为零的点集（如$D$中有限个孤立点，或者有限条光滑曲线）上，则$f(x,y)$在$D$上可积；
  2. 若$D$上使$f(x,y)\neq g(x,y)$的点分布在$D$中测度为零的点集上，则$f(x,y)$和$g(x,y)$在$D$上有相同的可积性。当它们可积时，有
    \[\int_D f=\int_D g \]
    也就是说，在测度为零的点集上任意改变函数的值，不改变函数的可积性，当函数可积时，也不改变积分值。

定理 5.
$D$是由有限条分段光滑曲线围成的区域，$f(x,y)$, $g(x,y)$是$D$上的可积函数。

  1. （线性）对任意常数$c_1$, $c_2$ 和$c_1f+c_2g$在$D$上可积，且

    \[\int_D(c_1f+c_2g)=c_1\int_Df+c_2\int_D g \]

  2. （乘积） $f(x,y)g(x,y)$在$D$上可积

  3. （保序性）若在$D$上$f(x,y)\geq g(x,y)$，则$\int_Df\geq\int_D g$

  4. （绝对可积性）$|f(x,y)|$在$D$上可积，且有

    \[|\int_D f|\leq \int_D|f| \]

定理 6.
$D$是两个可测点集$D_1$, $D_2$的拼接（即没有公共内点）， 若函数$h(x,y)$在$D_1$, $D_2$上都可积， 则$f(x,y)$在$D_1\cup D_2$上可积，且

\[\int_{D_1\cup D_2}f=\int_{D_1}f+\int_{D_2}f \]

定理 7. (积分中值定理)
$f(x,y)$在连通闭域$\bar D$中连续，则存在$(x_0,y_0)\in D$, 使得

\[\int_{D}f=f(x_0,y_0)\sigma(D) \]

二重积分的计算

先讨论矩形区域$D=[a,b]\times[c,d]$上的积分。

\begin{tikzpicture}[scale=1] %draw the axes \draw[->] (0,0,0) -- (3,0,0) node[anchor=west]{$y$}; \draw[->] (0,0,0) -- (0,2,0) node[anchor=south]{$z$}; \draw[->] (0,0,0) -- (0,0,3) node[anchor=east]{$x$}; % draw domain in plane Oxy \draw[thick] (0,0,0) -- (0,0,2.5) -- (2.5,0,2.5) -- (2.5,0,0) -- cycle; \draw[dashed, red] (0,0,2.0) -- (2.5,0,2.0); \draw[dashed, red] (0,0,1.5) -- (2.5,0,1.5); \draw[dashed, red] (0,0,1.0) -- (2.5,0,1.0); \draw[dashed, red] (0,0,0.5) -- (2.5,0,0.5); \draw[dashed, blue] (2.0,0,0) -- (2.0,0,2.5); \draw[dashed, blue] (1.5,0,0) -- (1.5,0,2.5); \draw[dashed, blue] (1.0,0,0) -- (1.0,0,2.5); \draw[dashed, blue] (0.5,0,0) -- (0.5,0,2.5); % draw the surface % == define coordinate matrix \foreach \x/\z in {0/1.75,1/1.83, 2/1.9, 3/1.85, 4/1.78, 5/1.8}{ \coordinate (p0\x) at (\x*0.5, \z, 0); } \foreach \x/\z in {0/1.70,1/1.83, 2/1.9, 3/1.85, 4/1.78, 5/1.88}{ \coordinate (p1\x) at (\x*0.5, \z, 0.5); } \foreach \x/\z in {0/1.55,1/1.83, 2/1.69, 3/1.85, 4/1.78, 5/1.68}{ \coordinate (p2\x) at (\x*0.5, \z, 1.0); } \foreach \x/\z in {0/1.4, 1/1.43, 2/1.5, 3/1.5, 4/1.53, 5/1.62}{ \coordinate (p3\x) at (\x*0.5, \z, 1.5); } \foreach \x/\z in { 0/1.28 , 1/1.32, 2/1.35, 3/1.32, 4/1.3, 5/1.43}{ \coordinate (p4\x) at (\x*0.5, \z, 2.0); } \foreach \x/\z in {0/1.25, 1/1.28, 2/1.35, 3/1.4, 4/1.39, 5/1.3}{ \coordinate (p5\x) at (\x*0.5, \z, 2.5); } % == draw surface border \draw[thick] plot[smooth] coordinates { (p00) (p10) (p20) (p30) (p40) (p50)}; \draw[thick] plot[smooth] coordinates { (p05) (p15) (p25) (p35) (p45) (p55)}; \draw[thick] plot[smooth] coordinates { (p00) (p01) (p02) (p03) (p04) (p05)}; \draw[thick] plot[smooth] coordinates { (p50) (p51) (p52) (p53) (p54) (p55)}; % == draw curve in surface \draw[dashed, red] plot[smooth] coordinates { (p40) (p41) (p42) (p43) (p44) (p45)}; \draw[dashed, red] plot[smooth] coordinates { (p30) (p31) (p32) (p33) (p34) (p35)}; \draw[dashed, red] plot[smooth] coordinates { (p20) (p21) (p22) (p23) (p24) (p25)}; \draw[dashed, red] plot[smooth] coordinates { (p10) (p11) (p12) (p13) (p14) (p15)}; \draw[dashed, blue] plot[smooth] coordinates { (p01) (p11) (p21) (p31) (p41) (p51)}; \draw[dashed, blue] plot[smooth] coordinates { (p02) (p12) (p22) (p32) (p42) (p52)}; \draw[dashed, blue] plot[smooth] coordinates { (p03) (p13) (p23) (p33) (p43) (p53)}; \draw[dashed, blue] plot[smooth] coordinates { (p04) (p14) (p24) (p34) (p44) (p54)}; \draw[thin] (p42) -- (1.0,0,2.0); \draw[thin] (p43) -- (1.5,0,2.0); \draw[thin] (p33) -- (1.5,0,1.5); \draw[thin] (p32) -- (1.0,0,1.5); \filldraw[fill=red!20, opacity=0.5] (p42)--(p43)--(p33)--(p32)--cycle; \filldraw[fill=yellow!20, opacity=0.5] (1,0,2)--(1.5,0,2)--(1.5,0,1.5)--(1,0,1.5)--cycle; \node at (0,0,2.0) [anchor=east] {$x_i$}; %\node at (2.0,0,0) [anchor=south] {$y_j$}; \node at (1.5,0,2.50) [anchor=north] {$y_j$}; \end{tikzpicture}

作$[a,b]$和$[c,d]$的分割：

\[T_x: a=x_0<x_1<\cdots<x_n=b \]
\[T_y: c=y_0<y_1<\cdots<y_m=d \]

就得到$D$的分割

\[T: D_{ij}=[x_{i-1},x_i]\times[y_{j-1},y_j] \]

显然$\sigma(D_{ij})=\Delta x_i\Delta y_j$, $\Delta x_i=x_i-x_{i-1}$, $\Delta y_j=y_j-y_{j-1}$

由

\[\mbox{diam} D_{ij}=\sqrt{(\Delta x_i)^2+(\Delta y_j)^2}<\Delta x_i+\Delta y_j \]

得到

\[|T_x|, |T_y|<|T|<|T_x|+|T_y| \]
  • 对任意$\xi_i\in[x_{i-1},x_i]$, $\eta_j\in[y_{j-1},y_j]$，在$D_{ij}$中有点$M_{ij}=(\xi_i, \eta_j)$。
  • 则函数$f(x,y)$在分割$T$下的一个Riemann和为
    \[S(T,M)=\sum_{i,j=1}^{n,m}f(M_{ij})\Delta x_i\Delta y_j \]
    若$f(x,y)$在$D$上可积，则上式Riemann和的极限就是积分$A$

通常，把积分$\displaystyle \iint_D f(x,y)d\sigma$ 又记为更常见的形式

\[\displaystyle \iint_D f(x,y)dxdy \]

$\color{red}\forall\epsilon>0$，由

\[A=\lim_{|T|\to0}\sum_{i,j=1}^{n,m}f(M_{ij})\Delta x_i\Delta y_j \]

存在$\color{red}\delta>0$，当$|T|<\delta$时，对任意$(\xi_i,\eta_j)\in D_{ij}$成立

\[{\color{blue}(*)} \quad A-\frac{\epsilon}2<\sum_{j=1}^m\Delta y_j\left[\sum_{i=1}^nf(\xi_i,\eta_j)\Delta x_i\right]<A+\frac{\epsilon}2 \]

注意到，对于$\eta_j\in[c,d]$， $\displaystyle\sum_{i=1}^nf(\xi_i,\eta_j)\Delta x_i$是 $f(x,\eta_j)$在$[a,b]$上的Riemann和。

若对每个$y$，$f(x,y)$作为$x$的函数在$[a,b]$上可积， 记积分值为$\phi(y)=\int_a^bf(x,y)dx$，则

\[\lim_{|T_x|\to0}\sum_{i=1}^nf(\xi_i,\eta_j)\Delta x_i=\int_a^b f(x,\eta_j)dx=\phi(\eta_j) \]

对${\color{blue}(*)}$式取极限$|T_x|\to0$，则有

\[{\color{red}A-\epsilon<}A-\frac{\epsilon}2 \leq {\color{red}\sum_{j=1}^m\phi(\eta_j)\Delta y_j} \leq A+\frac{\epsilon}2{\color{red}<A+\epsilon} \]

对$\color{red}|T_y|<\delta$成立。

因此，$\phi(y)$在$[c,d]$上可积，且积分为$A$。

定理 8. (Fubini定理)
函数$f(x,y)$在二维闭区间$D=[a,b]\times[c,d]$上可积。

  1. 如果对每个$y\in[c,d]$，$f(x,y)$作为$x$的函数在$[a,b]$上可积，记 $\phi(y)=\int_a^bf(x,y)dx$ 则$\phi(y)$在$[c,d]$上可积，且有
    \[\int_c^d\phi(y)dy=\int_c^d\left[\int_a^bf(x,y)dx\right]dy=\iint\limits_Rf(x,y)dxdy \]
  2. 如果对每个$x\in[a,b]$，$f(x,y)$作为$y$的函数在$[c,d]$上可积，记 $\psi(x)=\int_c^df(x,y)dy$ 则$\psi(x)$在$[a,b]$上可积，且有
    \[\int_a^b\psi(x)dx=\int_a^b\left[\int_c^df(x,y)dy\right]dx=\iint\limits_Rf(x,y)dxdy \]
  1. 若同时满足条件1和条件2，则
    \[\int_D f=\int_c^d\left[\int_a^bf(x,y)dx\right]dy =\int_a^b\left[\int_c^df(x,y)dy\right]dx \]

定理表明，二维区域上函数的二重积分，可化为先对一个变量的积分，再对另一个变量的积分。 这种积分过程称为累次积分。

几何上看，面包的体积，可以分为切片面包的体积来表示。

物理上看，薄板的质量，可以分为一些细长条来计算。

例 3. (例10.1.2) 求$\displaystyle\iint_D x\cos (xy) dxdy$， $D=[0,\pi]\times[0,1]$

例 4. 计算二重积分

\[\iint\limits_D\frac1{(x+y)^2}dxdy , D=[3,4]\times[1,2] \]

I型区域$D$是由曲线$y=\phi_1(x)$, $y=\phi_2(x)$和直线$x=a$, $x=b$围成 的区域，即

\[D=\{(x,y)|\phi_1(x)\leq y\leq\phi_2(x), a\leq x\leq b\} \]
\begin{tikzpicture}[scale=1.4] \draw[->] (-0.250,0) -- (3,0) node[anchor=west]{$x$}; \draw[->] (0,-0.20) -- (0,2.3) node[anchor=south]{$y$}; \node at (0,0) [anchor=north east] {$O$}; \coordinate (a1) at (0.5, 1.3); \coordinate (a2) at (0.5, 0.7); \coordinate (a3) at (0.5, 0.0); \coordinate (b1) at (2.65, 1.3); \coordinate (b2) at (2.65, 0.7); \coordinate (b3) at (2.65, 0.0); \draw[dashed] (a3) -- (a1); \draw[dashed] (b3) -- (b1); \draw (a1) .. controls (1.0, 2.0) .. (b1); \draw (a2) .. controls (1.0, 0.5) .. (b2); \node at (0.5, 0) [anchor=north] {$a$}; \node at (2.65, 0) [anchor=north] {$b$}; \node at (1.8, 1.0) {$D$}; \node at (1.50, 0.3) {$y=\phi_1(x)$}; \node at (2.0, 2.15) [anchor=north] {$y=\phi_2(x)$}; \end{tikzpicture}

定理 9.
I型区域

\[D=\{(x,y)|\phi_1(x)\leq y\leq\phi_2(x), a\leq x\leq b\} \]

其中$\phi_1(x)$, $\phi_2(x)$为连续函数。$f(x,y)$在$D$上可积， 且对于$\forall x\in[a,b]$，积分 $\displaystyle\int_{\phi_1(x)}^{\phi_2(x)}f(x,y)dy$ 存在，则

\[\iint\limits_Df(x,y)dxdy=\int_a^b\left[\int_{\phi_1(x)}^{\phi_2(x)}f(x,y)dy\right]dx \]

证明： 设$D\in[a,b]\times[c,d]$，

\begin{tikzpicture}[scale=1.4] \draw[->] (-0.50,0) -- (3,0) node[anchor=west]{$x$}; \draw[->] (0,-0.50) -- (0,2) node[anchor=south]{$y$}; \node at (0,0) [anchor=north east] {$O$}; \filldraw[fill=black!20, opacity=0.5] (0.5, 0.2)--(2.65, 0.2)--(2.65, 1.9)--(0.5, 1.9)--cycle; \node at ( 0, 0.2) [anchor=east] {$c$}; \node at ( 0, 1.9) [anchor=east] {$d$}; \draw[dashed] (0,0.2)--(0.5,0.2); \draw[dashed] (0,1.9)--(0.5,1.9); \coordinate (a1) at (0.5, 1.3); \coordinate (a2) at (0.5, 0.7); \coordinate (a3) at (0.5, 0.0); \coordinate (b1) at (2.65, 1.3); \coordinate (b2) at (2.65, 0.7); \coordinate (b3) at (2.65, 0.0); \draw[dashed] (a3) -- (a1); \draw[dashed] (b3) -- (b1); \draw (a1) .. controls (1.0, 2.0) .. (b1); \draw (a2) .. controls (1.0, 0.5) .. (b2); \node at (0.5, 0) [anchor=north] {$a$}; \node at (2.65, 0) [anchor=north] {$b$}; \node at (1.8, 1.0) {$D$}; \node at (1.50, 0.3) {$y=\phi_1(x)$}; \node at (2.0, 2.3) [anchor=north] {$y=\phi_2(x)$}; \end{tikzpicture}

令

\[f^*(x,y)=\begin{cases} f(x,y), &(x,y)\in D \\ 0, & (x,y)\notin D \end{cases} \]
\[\begin{aligned} \iint_{[a,b]\times[c,d]}f^*(x,y)dxdy =& \iint_{D}f^*(x,y)dxdy +\iint_{D^*}f^*(x,y)dxdy \\ =&\iint_D f(x,y)dxdy \end{aligned} \]

对$[a,b]$中任意固定值$x$，有

\[\begin{aligned} \int_c^d f^*(x,y)dy=&\int_a^{y_1(x)}f^*(x,y)dy+\int_{y_1(x)}^{y_2(x)}f^*(x,y)dy+\int_{y_2(x)}^bf^*(x,y)dy \\ =&\int_{y_1(x)}^{y_2(x)}f(x,y)dy \end{aligned} \]

定理 10.
II型区域

\[D=\{(x,y)|\psi_1(y)\leq x\leq\psi_2(y), c\leq y\leq d\} \]

其中$\psi_1(y)$, $\psi_2(y)$为连续函数。$f(x,y)$在$D$上可积， 且对于$\forall y\in[c,d]$，积分 $\displaystyle\int_{\psi_1(y)}^{\psi_2(y)}f(x,y)dx$ 存在，则

\[\iint\limits_Df(x,y)dxdy=\int_c^d\left[\int_{\psi_1(y)}^{\psi_2(y)}f(x,y)dx\right]dy \]

II型区域

\begin{tikzpicture}[scale=1.4] \draw[->] (-0.50,0) -- (3,0) node[anchor=west]{$x$}; \draw[->] (0,-0.50) -- (0,2) node[anchor=south]{$y$}; \node at (0,0) [anchor=north east] {$O$}; \coordinate (a1) at (2.8, 0.35); \coordinate (a2) at (1.15, 0.35); \coordinate (a3) at (0.0, 0.35); \coordinate (b1) at (2.65, 1.93); \coordinate (b2) at (1.145, 1.93); \coordinate (b3) at (0.0, 1.93); \draw[dashed] (a3) -- (a1); \draw[dashed] (b3) -- (b1); \draw (a1) .. controls (2.0, 1.4) .. (b1); \draw (a2) .. controls (1.5, 1.5) .. (b2); \node at (a3) [anchor=east] {$c$}; \node at (b3) [anchor=east] {$d$}; \node at (1.8, 1.0) {$D$}; \node at (1.25, 1.35) [anchor=east] {$x=\psi_1(y)$}; \node at (2.3, 1.3) [anchor=west] {$x=\psi_2(y)$}; \end{tikzpicture}

例 5. 计算累次积分

\[\int_0^1dx\int_x^{\sqrt x}\frac{\sin y}y dy \]

例 6. 画出计算积分区域，改写计算顺序

\[\int_0^2dy\int_{\frac{y}2}^yf(x,y)dx+\int_2^4dy\int_{\frac{y}2}^2f(x,y)dx \]
\begin{tikzpicture} \begin{axis}[domain=0:1,axis lines=middle,xlabel=$x$, ylabel=$y$, samples=40] \addplot[red!90!black] {sqrt(x)}; \addplot[red!90!black] {x}; \end{axis} \end{tikzpicture}
\begin{tikzpicture} \begin{axis}[axis lines=middle,xlabel=$x$, axis equal=true, %unit vector ratio=1 1, ylabel=$y$, samples=40, xmax=2.1, ymax=4.2] \addplot[domain=0:1, red!90!black] {2*x}; \addplot[domain=0:2, red!90!black] {x}; \addplot[domain=1:2, blue!90!black] {2*x}; \addplot[domain=2:4, blue!90!black] (2,x); \end{axis} \end{tikzpicture}

例 7. 画出计算积分区域，改写计算顺序

\[\int_{-1}^0dy\int_{-2\sqrt{y+1}}^{2\sqrt{y+1}}f(x,y)dx+\int_0^8dy\int_{-2\sqrt{1+y}}^{2-y}f(x,y)dx \]

例 8. (例10.1.8) 计算由两个圆柱面$x^2+y^2=a^2$与$x^2+z^2=a^2$所围成的立体的体积

\begin{tikzpicture} \begin{axis}[axis lines=middle,xlabel=$x$, axis equal=true, %unit vector ratio=1 1, ylabel=$y$, samples=40, xmax=2.1, ymax=8.2] \addplot[domain=-2:2, red!90!black] {x*x/4-1}; %\addplot[domain=0:2, red!90!black] {x}; \addplot[domain=-6:-2, blue!90!black] {x*x/4-1}; \addplot[domain=-6:2, blue!90!black] {2-x}; \end{axis} \end{tikzpicture}

谢谢

\begin{tikzpicture}[scale=1.4] \draw[->] (-0.50,0) -- (3,0) node[anchor=west]{$x$}; \draw[->] (0,-0.50) -- (0,2) node[anchor=south]{$y$}; \node at (0,0) [anchor=north east] {$O$}; \coordinate (a1) at (2.8, 0.35); \coordinate (a2) at (1.15, 0.35); \coordinate (a3) at (0.0, 0.35); \coordinate (b1) at (2.65, 1.93); \coordinate (b2) at (1.145, 1.93); \coordinate (b3) at (0.0, 1.93); \draw[dashed] (a3) -- (a1); \draw[dashed] (b3) -- (b1); \draw (a1) .. controls (2.0, 1.4) .. (b1); \draw (a2) .. controls (1.5, 1.5) .. (b2); \node at (a3) [anchor=east] {$c$}; \node at (b3) [anchor=east] {$d$}; \node at (1.8, 1.0) {$D$}; \node at (1.25, 1.35) [anchor=east] {$x=\psi_1(y)$}; \node at (2.3, 1.3) [anchor=west] {$x=\psi_2(y)$}; \end{tikzpicture}

在曲线上取两点$M$和$M'$，其横坐标分别为$x$与$x+dx$， 则两点的距离为

目录

本节读完

例 9. 谢

9. 图形可以放缩吗？

\begin{tikzpicture} %draw the axes \draw[->] (0,0,0) -- (3,0,0) node[anchor=west]{$y$}; \draw[->] (0,0,0) -- (0,2,0) node[anchor=west]{$z$}; \draw[->] (0,0,0) -- (0,0,3) node[anchor=west]{$x$}; \tikzset{ seagull/.pic={ \draw[dashed] (0.2, -0.4)--(1.4, -0.85)--(2.3,-0.25); \draw[dashed] (0.2, -0.4) .. controls (1,0) .. (2.3,-0.25); \draw[thick] (0.2, 1.4).. controls(0.75, 1.35) ..(1.4, 0.75); \draw[thick] (2.3, 1.25).. controls(1.7, 1.2)..(1.4, 0.75); \draw[thick] (0.2, 1.4) .. controls (1,2) .. (2.3,1.25); \draw[dashed] (0.2,1.4)--(0.2,-0.4); \draw[dashed] (2.3, 1.25)--(2.3, -0.25); \draw[dashed] (1.4, 0.75)--(1.4, -0.85); } } \draw (0,0) pic {seagull}; \draw (0.75,-0.231) pic [xscale=0.13] {seagull}; \end{tikzpicture}