∑是介于平面Z=0和Z=H（H>0）之间的圆柱面x^2+y^2=R^2,则∫∫1/(x^2+y^2)ds=?

Question

∑是介于平面Z=0和Z=H（H>0）之间的圆柱面x^2+y^2=R^2,则∫∫1/(x^2+y^2)ds=?

Answer 1

∫∫Σ 1/(x²+y²) dS
= ∫∫Σ 1/R² dS
= (1/R²)∫∫Σ dS
= (1/R²)(2πRH)
= 2π(H/R)
其中∫∫Σ dS是圆柱曲面面积