设Ω是由x^2+y^2≤8,x^2+2y^2≥z^2以及z≥0所确定的闭区域,试计算I=2zdxdyd三重积分

Question

设Ω是由x^2+y^2≤8,x^2+2y^2≥z^2以及z≥0所确定的闭区域,试计算I=2zdxdyd三重积分

Answer 1

z^2≤x^2+2y^2=r^2+r^2(sint)^2=r^2[1+(sint)^2]
I = ∫∫∫2zdxdydz
= ∫dt∫rdr∫2zdz
= ∫[1+(sint)^2]dt∫r^3dr
= ∫[3/2-(1/2)cos2t]dt[r^4/4]
= 16[3t/2-(1/4)sin2t] = 48π