x*2+y*2+z*2

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x*2+y*2+z*2x*2+y*2+z*2x*2+y*2+z*2球坐标变换:x=rsinacosbz=rsinasinby=rcosa∂(x,y,z)/∂(r,a,b)=r^

x*2+y*2+z*2
x*2+y*2+z*2

x*2+y*2+z*2
球坐标变换:
x=rsinacosb
z=rsinasinb
y=rcosa
∂(x,y,z)/∂(r,a,b)=r^2sina
所以∫∫∫(x^2+z^2)dxdydz
=∫[0,2π] db ∫[0,π] da ∫[0,1] r^2sin^2a*r^2*sina dr
=2π* ∫[0,π] sin^3a da ∫ [0,1]r^4 dr
∫[0,π] sin^3a da
= ∫[0,π] (1-cos^2a)*sina da
=- ∫[0,π] (1-cos^2a) d(cosa)
= -(cosa-1/3cos^3a) [0,π]
=4/3
∫ [0,1]r^4 dr=1/5r^5 [0,1]=1/5
所以∫∫∫(x^2+z^2)dxdydz
=2π* ∫[0,π] sin^4a da ∫ [0,1]r^4 dr
=2π*4/3*1/5
=8π/15

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