Question

Use Green’s Theorem to evaluate the line integral along the given positively oriented curve. ∫c cos y dx + x^2 sin y dy, C is the rectangle with vertices (0, 0), (5, 0), (5, 2), and (0, 2)

Answer 1

By Green's theorem, we have

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

where $C$ is the *boundary* of the rectangle $R$.

The integral is then

$\displaystyle\int_0^5\int_0^2(2x+1)\sin y\,\mathrm dy\,\mathrm dx$

$=\displaystyle(\cos0-\cos2)\int_0^5(2x+1)\,\mathrm dx$

$=\displaystyle(1-\cos2)(5^2+5)=\boxed{30(1-\cos 2)}$