Question

Use Green's Theorem to evaluate F · dr. C (Check the orientation of the curve before applying the theorem.)F(x, y) = y cos(x) − xy sin(x), xy + x cos(x) , C is the triangle from (0, 0) to (0, 8) to (2, 0) to (0, 0)

Answer 1

Notice that C has a clockwise orientation. By Green's theorem, we have

$\displaystyle\int_C\mathbf F(x,y)\cdot\mathrm d\mathbf r=-\iint_D\left(\frac{\partial(xy+x\cos x)}{\partial x}-\frac{\partial(y\cos x-xy)}{\partial y}\right)\,\mathrm dx\,\mathrm dy$

where D is the triangule region with C as its boundary, given by the set

$D=\{(x,y)\mid0\le x\le2\land0\le y\le8-4x\}$

So we have

$\displaystyle\int_C\mathbf F(x,y)\cdot\mathrm d\mathbf r=-\int_0^2\int_0^{8-4x}((y+\cos x-x\sin x)-(\cos x-x\sin x))\,\mathrm dy\,\mathrm dx$

$\displaystyle\int_C\mathbf F(x,y)\cdot\mathrm d\mathbf r=-\int_0^2\int_0^{8-4x}y\,\mathrm dy\,\mathrm dx=\boxed{-\dfrac{64}3}$