设Ω={（x，y，z）|x2+y2≤z≤1}，则Ω的形心的竖坐标.z=2323．

Question

设Ω={（x，y，z）|x²+y²≤z≤1}，则Ω的形心的竖坐标

.

z

=

(2)/(3)

(2)/(3)

．

Answer 1

∵Ω的体积V＝

∭

Ω

dxdydz
=

∫

1 0

dz

∬

x2+y2≤z

dxdy
=

∫

1 0

πzdz=(π)/(2)
∴

.

z

＝(∭Ωzρdxdydz)/(M)＝(1)/(V)

∭

Ω

zdxdydz
=

∫

1 0

zdz

∬

x2+y2≤z

dxdy
=(1)/(V)

∫

1 0

πz2dz
=(1)/(V)•(π)/(3)
=(2)/(3)