Answer :

Solution :

Green’s theorem, ∮_c∅dx+ψdy=∬s(∂ψ/∂x-∂∅/∂y) dxdy

Checking from options,

From option (A) :

∮_cxdy=∬_s(1-0)dxdy=∬_sdxdy            ( ∅=0,ψ=x)     (Area of region R in anticlockwise direction)

From option (B) :

1/2∮_c(xdy-ydy)= 1/2 ∬_s(1-(-1))dxdy         ( ∅=-y,ψ=x)

                             =1/2 ∬_s2dxdy = ∬_sdxdy        [Area of region R in anticlockwise direction]

From option (C) :

∮_cdxdy= Area of region R

From option (D) :

∮_cydx=∬_s(0-(1))dxdy                ( ∅=y,ψ=0)

           =∬-1dxdy= -∬dxdy        [Area of region R in clockwise direction]

Hence, the correct option is (D).