Question

计算三重积分∫〈0—1〉dx∫〈0—1－x〉dy∫〈0—y/2〉［cos(z)/(2z－1)^2］dz的值

Answer 1

交换积分次序为先x再y最后为z的三次积分：

∫〈0—1〉dx∫〈0—1－x〉dy∫〈0—y/2〉［cos(z)/(2z－1)^2］dz

＝∫〈0—1/2〉dz∫〈2z—1〉dy∫〈0—1-y〉［cos(z)/(2z－1)^2］dz

＝∫〈0—1/2〉[cos(z)/(2z－1)^2］dz∫〈2z—1〉(1-y）dy

＝∫〈0—1/2〉(1/2)cos(z)dz=(1/2)sin(1/2)