计算三重积分 其中Ω为球体

2025-05-16 07:31:05
推荐回答(1个)
回答1:

用球坐标变换
x=rsinφcosθ,y=rsinφsinθ,z=rcosφ,0<=r<=a,0<=φ<=π,0<=θ<=2π

原式=∫(0,2π)dθ∫(0,π)dφ∫(0,a)r^2*r^2*sinφdr
=2π*∫(0,π)sinφdφ*∫(0,a)r^4dr
=(4π/5)*r^5|(0,a)
=(4π/5)*a^5