We need to evaluate ∮CFds, where F=(P(x,y),Q(x,y))=(x2y,−xy2), and C is the contour of the circle of radius 2, centered at the origin.
According to Green's theorem, ∮CFds=∬D(∂x∂Q−∂y∂P)dA=∬(−y2−x2)dA=−∬(x2+y2)dA , where we need to take the surface integral inside the circle.
Let us use polar coordinates: x=rcosθ,y=rsinθ,dA=rdrdθ , 0≤r≤2,0≤θ≤2π. Hence, the integrand is −∫02πdθ∫02r⋅r2dr=−2π4r4∣02=−8π.
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