Question

Let F⃗ =6(x+y)i⃗ +sin(y)j⃗ . Find the line integral of F⃗ around the perimeter of the rectangle with corners (4,0), (4,7), (−2,7), (−2,0), traversed in that order.
line integral/ greens theorum question

Answer 1

\(\vec{F}=(6(x+y),\sin y)=6(x+y)\vec{i}+\sin y\vec{j},\) so by Green's Theorem you have
\[\begin{align*}\oint_C\big(6(x+y)~dx+\sin y~dy\big) &= \int\int_D\left(\frac{\partial}{\partial x}\sin y-\frac{\partial}{\partial y}6(x+y)\right)~dx~dy\\
&=\int_0^7\int_{-2}^4\left(0-6\right)~dx~dy
\end{align*}\]