Question

求这道题的过程有关高等数学二重积分问题

求这道题详细过程

Answer 1

∫∫D f(x，y)dxdy
=∫∫D1 x²dxdy＋∫∫D2 1/∨(x²＋y²)dxdy
=4∫(－1，0) x²dx∫(0，x＋1)dy＋4∫(0，π/2) dθ∫[1/(cosθ＋sinθ)，2/(cosθ＋sinθ)] dr
=4∫(－1，0) (x³＋x²)dx＋4∫(0，π/2) 1/(sinθ＋cosθ)dθ
=(x^4＋4/3 x³)|(－1，0)＋∨2 ln{[1-cos(θ+π/4)]/[1+cos(θ+π/4)]} |(0，π/2)
=1/3＋∨2｛ ln[(2＋∨2)/(2－∨2)]－ln[(2－∨2)/(2＋∨2)]｝
=1/3 ＋∨2 ln[(6＋4∨2)/(6－4∨2)]
=1/3＋∨2 ln(17＋12∨2)