Question

The line integral of xydx×x^2y^3dy where c is the triangle with points (0,0) (1,0) (1,2) using green's theorem

Answer 1

If $\mathcal C$ is the boundary of the triangle $D$, then by Green's theorem

$\displaystyle\int_{\mathcal C}xy\,\mathrm dx+x^2y^3\,\mathrm dy=\iint_D\left(\frac{\partial(x^2y^3)}{\partial x}-\frac{\partial(xy)}{\partial y}\right)\,\mathrm dA$
$=\displaystyle\int_{x=0}^{x=1}\int_{y=0}^{y=2x}(2xy^3-x)\,\mathrm dy\,\mathrm dx$
$=\displaystyle\int_{x=0}^{x=1}x\int_{y=0}^{y=2x}(2y^3-1)\,\mathrm dy\,\mathrm dx$
$=\displaystyle\int_{x=0}^{x=1}x(8x^4-2x)\,\mathrm dx=\frac23$