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∫Pdx + ∫Qdy = ∫∫(dQ/dx)-(dP/dy)

A = ∫xdy = -∫ydx = ½*∫xdy - ydx

I know that the mass of a region D with constant density function is ∫kdA (which is the area times some constant K). Let's make it easy and assume that k = 1 with the area A. So, the centroid of the region D would be located at (1/A*∫∫xdA) and (1/A*∫∫ydA). So, if I set Pdx as -ydx, and Qdy as xdy, I would get from Green's Theorem that ∫Pdx = -∫∫ydxdy and ∫Qdy = ∫∫xdxdy. But if you divide by A this is the expression for the coordinates of the centroid, since ∫∫xdxdy = ∫∫xdA and -∫∫ydxdy = ∫∫ydA. So you have the coordinates as 1/2A∫x^2/dy and 1/2A∫-y^2dx as the coordinates? Thanks.