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((\partial(x^2\sin y))/(\partial x)-(\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}`