x=rcos φ
y=rsin φ
(x+y)^3=(rcos φ+rsin φ)^3=r^3(cos φ +sin φ )^3
xy=rcos φ *rsinφ =r^2sinφcosφ
Уравнение петли:
r^3(cos φ +sin φ )^3=r^2sinφcosφ
r=sinφcosφ/(sin φ+cos φ)^3
0 ≤r ≤ sinφcosφ/(sin φ+cos φ)^3
0 ≤ φ ≤ π/2
S= ∫ ∫ _(D)dxdy= ∫^(π/2)_(0)dφ ∫ ^(sinφcosφ/(sin φ+cos φ)^3)_(0) rdr=
= ∫^(π/2)_(0) (r^2/2)|^(sinφcosφ/(sin φ+cos φ)^3)_(0) d φ=
=(1/2) ∫^(π/2)_(0)sin^2 φ cos^2 φdφ/(sin φ +cos φ )^6
По формулам тригонометрии:
sin^2 φ cos^2 φ=(1/4)sin^2 2φ
(sin φ +cos φ )^6=((sin φ +cos φ )^2)^3=(sin^2φ +2sinφcos φ+cos^2 φ)^3= (1+sin2 φ)^3
получаем
(1/8) ∫^(π/2)_(0)sin^22φ dφ/(1+sin2φ)^3