Answer
$\dfrac{-1}{12}$
Work Step by Step
Green's Theorem states that:
$\oint_CP\,dx+Q\,dy=\iint_{D}(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y})dA$
Work done:$\int_{C} F \cdot dr=\int_{C} x(x+y) dx+xy^2 dy$
We set up the line integral and find out the integrand of the double integral as follows:
$\oint_CP\,dx+Q\,dy= \iint_{D}(\dfrac{\partial (xy^2)}{\partial x}-\dfrac{\partial (x^2+xy)}{\partial y})dA$
or, $=\iint_{D} y^2-x dA$
or, $= \int_{0}^{1} \int_{0}^{1-x} (y^2-x) \ dy \ dx $
or, $= \int_{0}^{1} \dfrac{(1-x)^3}{3}-x+x^2 dx$
or, $= [\dfrac{-(1-x)^4}{12}-\dfrac{x^2}{2}+\dfrac{x^3}{3}]_0^1$
or, $=\dfrac{-1}{12}$
