定义
在有界闭区域D上，f(x,y)在D上有界
将D划分为若干小区<span class="equation-text" data-index="0" data-equation="\Delta a_i" contenteditable="false"><span></span><span></span></span><br>
对于<span class="equation-text" data-index="0" data-equation="\forall (\zeta_i,\eta_i)\in\Delta_i" contenteditable="false"><span></span><span></span></span>作<span class="equation-text" data-index="1" data-equation="\sum\limits_{i=1}^{n}f(\zeta_i,\eta_i)\Delta a_i" contenteditable="false"><span></span><span></span></span><br>
取\lambda为每个小区域\Delta a_i直径之最大者，<br>若<span class="equation-text" data-index="0" data-equation="\lim\limits_{\lambda\to0}\sum\limits_{i=1}^{n}f(\zeta_i,\eta_i)\Delta a_i" contenteditable="false"><span></span><span></span></span>存在，
称f(x,y)在D上可积，称此极限为其在D上的二重积分，记为<span class="equation-text" data-index="0" data-equation="\iint\limits_D f(x,y)\mathrm da" contenteditable="false"><span></span><span></span></span><br>
Note
<span class="equation-text" data-index="0" data-equation="\lim\limits_{\lambda\to0}\sum\limits_{i=1}^{n}f(\zeta_i,\eta_i)\Delta a_i" contenteditable="false"><span></span><span></span></span>与D划分方法、<span class="equation-text" data-index="1" data-equation="f(\zeta_i,\eta_i)" contenteditable="false"><span></span><span></span></span>取法无关<br>
f(x,y)在单位正方形区域D(0~1,0~1)上可积，<br>将积分区域在 xy轴上分别划分为m,n份，则<br>积分区域面积为<span class="equation-text" data-index="0" data-equation="\frac 1{mn}" contenteditable="false"><span></span><span></span></span>，<span class="equation-text" data-index="1" data-equation="f(\zeta_i,\eta_i)=f(\frac i m,\frac j n)" contenteditable="false"><span></span><span></span></span><br>
例：求<span class="equation-text" data-index="0" data-equation="\lim\limits_{\tiny\begin{aligned}m\to\infin\\n\to\infin\end{aligned}}\sum\limits_{i=1}^m\sum\limits_{j=1}^n\frac{n}{(m+i)(n^2+j^2)}" contenteditable="false"><span></span><span></span></span><br>
性质<br>
<span class="equation-text" data-index="0" data-equation="\iint f(x,y)+g(x,y)\mathrm da=\iint f(x,y)\mathrm da+\iint g(x,y)\mathrm da" contenteditable="false"><span></span><span></span></span>
常数可提取<br>
若<span class="equation-text" data-index="0" data-equation="D=D_1+D_2,且D_1\cap D_2=\empty" contenteditable="false"><span></span><span></span></span>，则<span class="equation-text" data-index="1" data-equation="\iint\limits_D=\iint\limits_{D_1}+\iint\limits_{D_2}" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\iint\limits_D 1da=A" contenteditable="false"><span></span><span></span></span>
有界闭区域D，<span class="equation-text" data-index="0" data-equation="f(x,y)\in C(D)" contenteditable="false"><span></span><span></span></span>，则<span class="equation-text" data-index="1" data-equation="\exist (\zeta,\eta)\in D" contenteditable="false"><span></span><span></span></span>，使 <span class="equation-text" data-index="2" data-equation="\iint\limits_D f(x,y)da=f(\zeta,\eta)A" contenteditable="false"><span></span><span></span></span>
对称性<br>
D关于任意轴对称
f()关于该轴
“奇函数”<br>
<span class="equation-text" data-index="0" data-equation="\iint\limits_D f(x,y)da=0" contenteditable="false"><span></span><span></span></span>
“偶函数”
<span class="equation-text" data-index="0" data-equation="\iint\limits_D f(x,y)da=\iint\limits_{D_1} f(x,y)da" contenteditable="false"><span></span><span></span></span>
D关于y=x对称<br>
<span class="equation-text" data-index="0" data-equation="\iint\limits_D f(x,y)da=\iint\limits_D f(y,x)da" contenteditable="false"><span></span><span></span></span>
例：求<span class="equation-text" data-index="0" data-equation="I=\iint\limits_D\frac x {x+y}\mathrm da" contenteditable="false"><span></span><span></span></span><br>
积分法<br>
直角坐标<br>
积分次序
<span class="equation-text" data-index="0" data-equation="\iint\limits_Df(x,y)\mathrm da=\int_a^b\mathrm dx\int_{\varphi_1(x)}^{\varphi_2(x)}f(x,y)\mathrm dy" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="D=\{(x,y)|a\leq x\leq b,\\\varphi_1(x)\leq y\leq\varphi_2(x)\}" contenteditable="false"><span></span><span></span></span><br>X型区域<br>
并非两积分相乘，而是嵌套
<span class="equation-text" data-index="0" data-equation="\iint\limits_Df(x,y)\mathrm da=\int_c^d\mathrm dy\int_{\varphi_1(y)}^{\varphi_2(y)}f(x,y)\mathrm dx" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="D=\{(x,y)|c\leq y\leq d,\\\varphi_1(y)\leq x\leq \varphi_2(y)\}" contenteditable="false"><span></span><span></span></span><br>Y型区域<br>
实践<br>
作图，确定积分区域
凡是X型可算，Y型也可算，结果必定一样
XY型运算量随被积函数不同而有差距
不可积函数
<span class="equation-text" data-index="0" data-equation="x^{2n}e^{\pm x^2}\mathrm dx" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="e^{\frac k x}\mathrm dx" contenteditable="false"><span></span><span></span></span>
<span class="equation-text" data-index="0" data-equation="\cos \frac k x sin \frac k x \mathrm dx" contenteditable="false"><span></span><span></span></span>
例：<span class="equation-text" data-index="0" data-equation="\int_0^1\mathrm dy\int_y^1 e^{x^2}dx" contenteditable="false"><span></span><span></span></span>
极坐标
识别：积分区域边界或f(x,y)含有<span class="equation-text" data-index="0" data-equation="x^2+y^2" contenteditable="false"><span></span><span></span></span>
令<span class="equation-text" data-index="0" data-equation="\left\{\begin{aligned}x=r\cos\theta\\y=r\sin\theta\end{aligned}\right." contenteditable="false"><span></span><span></span></span>，有<span class="equation-text" data-index="1" data-equation="D=\{(x,y)|\alpha\leq\theta\leq\beta,r_1(\theta)\leq r\leq r_2(\theta)\}" contenteditable="false"><span></span><span></span></span><br>且另有<span class="equation-text" data-index="2" data-equation="\mathrm da=r\mathrm dr\mathrm d\theta" contenteditable="false"><span></span><span></span></span>，故<span class="equation-text" data-index="3" data-equation="\iint\limits_D f(x,y)\mathrm da=\int_{\alpha}^{\beta}\mathrm d\theta\int_{r_1(\theta)}^{r_2(\theta)}{\color{red}r}f(r\cos\theta,r\sin\theta)\mathrm dr" contenteditable="false"><span></span><span></span></span><br>
实践<br>
即便是极坐标也可以利用对称性和<br>被积函数奇偶性简化被积函数<br>
例：计算<span class="equation-text" data-index="0" data-equation="I=\iint_D(x^3+2x^2y)\mathrm da" contenteditable="false"><span></span><span></span></span>
例：计算<span class="equation-text" data-index="0" data-equation="I=\int_D(x^2+xy)\mathrm da" contenteditable="false"><span></span><span></span></span>，其中x,y满足<span class="equation-text" data-index="1" data-equation="x^2+y^2\leq2x" contenteditable="false"><span></span><span></span></span><br>
