交换积分次序的基本具体步骤如下:
1、对于二重积分,如果x和y的积分上下限都为负无穷和正无穷,那么直接调换dx,dy即可,如下图所示。
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/314e251f95cad1c89649b1466f3e6709c83d51e4?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
2、对于更一般的二重积分,首先需要根据积分式画出积分区域,上下限都为常数时,画出的积分区域是矩形。
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/7e3e6709c93d70cf0c08e76ce8dcd100bba12be4?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
3、这样在交换dx和dy的同时,交换积分符号,如下图所示。
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/c83d70cf3bc79f3dc113a951aaa1cd11738b29e4?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
4、很多二重积分的上下限是x或者y的函数,这时也要先画出积分区域,如下图。
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/71cf3bc79f3df8dcebcfc2f0dd11728b461028e4?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
5、为了先对y积分,在坐标系中画一条x轴的平行线,如下。
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/bba1cd11728b471086ddb0add3cec3fdfd0323e4?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
6、然后不断移动这条平行线,先写出y的上下限x2和x1,然后根据平行线,写出x的上下限x2,x。如下图所示。
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/0e2442a7d933c895b38cd51ac11373f0830200e4?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)
7、对于三重积分,其交换积分顺序的基本思想相同,可以利用数形结合的方法来处理,如下。
![](https://video.ask-data.xyz/img.php?b=https://iknow-pic.cdn.bcebos.com/faedab64034f78f099f9d79269310a55b2191ce4?x-bce-process=image%2Fresize%2Cm_lfit%2Cw_600%2Ch_800%2Climit_1%2Fquality%2Cq_85%2Fformat%2Cf_auto)