Question

Use Green's Theorem to evaluate the line integral along the given positively oriented curve. C cos(y) dx + x2 sin(y) dy C is the rectangle with vertices (0, 0), (5, 0), (5, 4), (0, 4)

Answer 1

By Green's theorem,

$\displaystyle\int_C\cos y\,\mathrm dx+x^2\sin y\,\mathrm dy=\iint_D\left(\frac{\partial(x^2\sin y)}{\partial x}-\frac{\partial(\cos y)}{\partial y}\right)\,\mathrm dx\,\mathrm dy$

where $D$ is the region with boundary $C$, so we have

$\displaystyle\iint_D(2x+1)\sin y\,\mathrm dx\,\mathrm dy=\int_0^5\int_0^4(2x+1)\sin y\,\mathrm dy\,\mathrm dx=\boxed{60\sin^22}$