二重积分

一元函数的积分可以表示为平面图形的面积
二元函数的二重积分可以表示为空间曲面图形的体积；

一，概念

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-1.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-2.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-3.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-4.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-5.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-6.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-7.png" width="500" />

二，二重积分的计算

• 直角坐标系下计算，对谁积分谁就是变量，其余的都看为常数，如果即是上线结构又是左右结构就看被积函数关于x的表达式和关于y的表达式，哪一个简单就先对谁积分；

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-8.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-9.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-10.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-11.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-12.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-13.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-14.png" width="500" />
<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-15.png" width="500" />

• 极坐标系下计算( 在积分区域D为圆，半圆，圆环，扇形或被积函数为f(x^2+y2)的形式利用极坐标计算二重积分）
  <img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-16.png" width="500" />
  <img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-17.png" width="500" />
  <img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-18.png" width="500" />
  <img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-19.png" width="500" />
  <img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-20.png" width="500" />
  <img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-21.png" width="500" />

二，二重积分的应用

计算曲面图形的体积

<img src="https://raw.githubusercontent.com/liangxifeng833/my_program/master/images/math/mult_weifen/two-jifen-22.png" width="500" />