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(\partial(xy+x\cos x))/(\partial x)-(\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}`