If [tex]\mathcal C[/tex] is the boundary of the triangle [tex]D[/tex], then by Green's theorem
[tex]\displaystyle\int_{\mathcal C}xy\,\mathrm dx+x^2y^3\,\mathrm dy=\iint_D\left(\frac{\partial(x^2y^3)}{\partial x}-\frac{\partial(xy)}{\partial y}\right)\,\mathrm dA[/tex]
[tex]=\displaystyle\int_{x=0}^{x=1}\int_{y=0}^{y=2x}(2xy^3-x)\,\mathrm dy\,\mathrm dx[/tex]
[tex]=\displaystyle\int_{x=0}^{x=1}x\int_{y=0}^{y=2x}(2y^3-1)\,\mathrm dy\,\mathrm dx[/tex]
[tex]=\displaystyle\int_{x=0}^{x=1}x(8x^4-2x)\,\mathrm dx=\frac23[/tex]
