学习一下用Mathematica计算三重积分的方法。
工具/原料
电脑
Mathematica
方法/步骤
1、先来画图：RegionPlot3D[x^2+y^2<=z<=1,{x,-1,1},{y,-1,1},{z,-0.5,1.5}
2、求zhege图形的体积：Integrate[Boole[x^2+y^2<=z<=1],{x,-Infinity,Infinity},{y,-Infinity,Infinity},{z,-Infinity,Infinity}]
3、如果每一个点的密度都是该点的竖坐标，求这个“物体”的质量：Integrate[zBoole[x^2+y^2<=z<=1],{x,-Infinity,Infinity},{y,-Infinity,Infinity},{z,-Infinity,Infinity}]
4、计算三重积分，题目如下图。Integrate[(x^2+y^2)Boole[x^2+y^2<=4&&Sqrt[x^2+y^2]<=z<=2],{x,-Infinity,Infinity},{y,-Infinity,Infinity},{z,-Infinity,Infinity}]
5、对Sqrt[x^2+y^2+z^2]求积分，积分区域鋈守踬痊是x^2+y^2+z^2<=z。直接积分：Integrate[Sqrt[x^2+y^2+z^2]Boole[x^2+y^2+z^2&造婷用痃lt;=z],{x,-Infinity,Infinity},{y,-Infinity,Infinity},{z,-Infinity,Infinity}]
6、转而尝试球坐标变换：令：{x->rSin[u]Cos[v],y->rSi艘绒庳焰n[u]Sin[v],z->鸡堕樱陨rCos[u]}于是有：dxdydz->(r^2)Sin[u]drdudv原题可以化为：Integrate[r^2Sin[u]Sqrt[x^2+y^2+z^2]/.{x->rSin[u]Cos[v],y->rSin[u]Sin[v],z->rCos[u]},{v,0,2Pi},{u,0,Pi/2},{r,0,Cos[u]}]
