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((\partial(x^2y^3))/(\partial x)-(\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`