Question

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, 12) to (3, 0) to (0, 0)

Answer 1

$C$ has a clockwise orientation, so in order to use Green's theorem (which is probably the "theorem" mentioned here), we have

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

where $D$ is the triangle region.

$=\displaystyle\int_0^3\int_0^{12-4x}(\cos x-x\sin x)-(y+(\cos x-x\sin x))\,\mathrm dy\,\mathrm dx$

$=\displaystyle-\int_0^3\int_0^{12-4x}y\,\mathrm dy\,\mathrm dx=\boxed{-72}$