求解一道三重积分∫[0→π/2] cosθsinθ dθ∫[0→π/4] sin³φdφ∫[0→2] r^4 dr
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求解一道三重积分∫[0→π/2]cosθsinθdθ∫[0→π/4]sin³φdφ∫[0→2]r^4dr求解一道三重积分∫[0→π/2]cosθsinθdθ∫[0→π/4]sin³
求解一道三重积分∫[0→π/2] cosθsinθ dθ∫[0→π/4] sin³φdφ∫[0→2] r^4 dr
求解一道三重积分
∫[0→π/2] cosθsinθ dθ∫[0→π/4] sin³φdφ∫[0→2] r^4 dr
求解一道三重积分∫[0→π/2] cosθsinθ dθ∫[0→π/4] sin³φdφ∫[0→2] r^4 dr
这个直接三个分开积啊
=∫[0→π/2] (1/2)sin(2θ)dθ∫[0→π/4] sin^2 φ d(-cosφ) (r^5/5)|[0→2]
=(1/4)(-cos(2θ))|[0→π/2]
*∫[0→π/4] (1-cos^2 φ) d(-cosφ)
*(2^5/5)
=(1/4)*(-1-1)*-(cosφ-(cos^3 φ)/3)|[0→π/4]
*32/5
=16/5*[(1-根号2/2)-(1-(根号/2)^3)/3]
=16/5*(2/3-5根号2/12)
=32/15-(4根号2)/3