Answer
$\dfrac{-1}{12}$
Work Step by Step
Green's Theorem states that:
$\oint_C M \,dx+ N \,dy=\iint_{D}(\dfrac{\partial N}{\partial x}-\dfrac{\partial M}{\partial y})dA$
We will use polar coordinates as follows: $x=r \cos \theta; y= r \sin \theta$
We need to work out with the line integral and evaluate the integrand of the double integral as follows:
Work done can be computed as:$\int_{C} F \cdot dr=\int_{C} x(x+y) dx+xy^2 dy$
$\oint_CP\,dx+Q\,dy= \iint_{D}(\dfrac{\partial (xy^2)}{\partial x}-\dfrac{\partial (x^2+xy)}{\partial y})dA \\=\iint_{D} y^2-x \ dA \\= \int_{0}^{1} \int_{0}^{1-x} (y^2-x) \ dy \ dx \\= \int_{0}^{1} \dfrac{1}{3} \times (1-x)^3-x+x^2 \ dx \\= [\dfrac{-(1-x)^4}{12}-\dfrac{x^2}{2}+\dfrac{x^3}{3}]_0^1 \\=\dfrac{-1}{12}$
