Answer
$$\dfrac{-1}{12}$$
Work Step by Step
Work done can be written as: $\int_{C} F \cdot dr=\int_{C} x(x+y) dx+xy^2 dy$
Green's Theorem states that:
$\oint_C A\,dx+B \,dy=\iint_{D}(\dfrac{\partial B}{\partial x}-\dfrac{\partial A}{\partial y}) dA $
We need to set up the line integral and compute the integrand of the double integral as follows:
$$ \ Work \ done: \int_{C} F \cdot dr=\int_{C} x(x+y) dx+xy^2 dy\\ = \iint_{D}(\dfrac{\partial (xy^2)}{\partial x}-\dfrac{\partial (x^2+xy)}{\partial y})dA \\=\iint_{D} y^2-x dA \\= \int_{0}^{1} \int_{0}^{1-x} (y^2-x) \ dy \ dx \\= \int_{0}^{1} \dfrac{(1-x)^3}{3}-x+x^2 dx \\= [\dfrac{-(1-x)^4}{12}-\dfrac{x^2}{2}+\dfrac{x^3}{3}]_0^1 \\=\dfrac{-1}{12}$$
