高等数学三重积分计算问题

2025-05-09 15:10:58
推荐回答(1个)
回答1:

(1) ∫∫∫ xdxdydz = ∫<-4,0>dx∫<-4-x,0>dy∫<0,x+y+4> xdz
= ∫<-4,0>xdx∫<-4-x,0>(x+y+4)dy = ∫<-4,0>xdx [(x+4)y+y^2/2]<-4-x,0>
= ∫<-4,0>xdx [(x+4)y+y^2/2]<-4-x,0> = (1/2)∫<-4,0> x(x+4)^2dx
= (1/2)∫<-4,0> (x^3+8x^2+16x)dx =(1/2)[x^4/4+(8/3)x^3+8x^2]<-4,0> =-32/3.
(2)∫∫∫zdzdydz = ∫<0,2π>dt∫<1,2>rdr∫ zdz
= ∫<0,2π>dt∫<1,2>r(8-r^4/2)dr = 2π[4r^2-r^6/12]<1,2> = 27/4.