Question

求三重积分∫∫∫x^2dxdydz D{(x,y,z) | x^2+y^2+z^2≤4，x^2+y^2+z^2≤4x}

Answer 1

设y=rcosa,z=rsina,则dydz=rdrda,

D={(x,y,z) | x^2+y^2+z^2≤4，x^2+y^2+z^2≤4x},是被平面x=1分成的两个球缺，

∴∫∫∫x^2dxdydz=∫da[∫x^2dx∫rdr+∫x^2dx∫rdr]

=π[∫x^2*(4x-x^2)dx+∫x^2*(4-x^2)dx]

=π[1-1/5+28/3-31/5]

=59π/15.

仅供参考.