用二重积分算体积v是球体x^2+y^2+z^2
1个回答
先计算两球的交线,易得：交线为z=r/2,平面z=r/2将这个公共部分分为两部分,这两部分是对称的,因此我们只求上半部分,然后2倍即可.
将z=r/2代入球面方程得：x²+y²=3r²/4
因此本题转化为计算球面x²+y²+z²=r²被圆柱面x²+y²=3r²/4截出的球冠体积,然后2倍.
球面x²+y²+z²=r²方程化为：z=√(r²-x²-y²)
A1=∫∫√(r²-x²-y²)dxdy 积分区域为：x²+y²≤3r²/4
用极坐标
=∫∫ρ√(r²-ρ²)dρdθ
=∫[0---->2π]dθ∫[0---->√3r/2] ρ√(r²-ρ²)dρ
=2π∫[0---->√3r/2] ρ√(r²-ρ²)dρ
=π∫[0---->√3r/2] √(r²-ρ²)d(ρ²)
=-(2/3)π(r²-ρ²)^(3/2) |[0---->√3r/2]
=(2/3)π[r³-(r²-3r²/4)^(3/2)]
=(2/3)π*(7/8)r³
=(7/12)πr³
A=2A1=(7/6)πr³
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