∫∫ln(x^2+y^2)dxdy,其中1

Question

∫∫ln(x^2+y^2)dxdy,其中1<=x^2+y^2<=9

Answer 1

∫∫ln(x^2+y^2)dxdy
=∫[0,2π]dθ ∫[1,3] ln(r^2)*rdr
= 2π * { 1/2 ∫[1,3] ln(r^2)dr^2 ｝
= π*∫[1,9] lntdt
= π*{tlnt|[1,9] - ∫[1,9] 1dt }
= π*{9ln9 -8}
= π*{18ln3 - 8 }