Question

Use Green's Theorem to find the work done by the force F(x, y) = x(x + y) i + xy2 j in moving a particle from the origin along the x-axis to (6, 0), then along the line segment to (0, 6), and then back to the origin along the y-axis.

Answer 1

$x+y=6$ is the equation of the line through the points (6, 0) and (0, 6).

By Green's theorem,

$\displaystyle\int_C\vec F(x,y)\cdot\mathrm d\vec r=\iint_D\left(\frac{\partial(xy^2)}{\partial x}-\frac{\partial(x(x+y))}{\partial y}\right)\,\mathrm dA$

where $C$ is the given path and $D$ is the triangular region with $C$ as its boundary. We have

$\displaystyle\iint_D(y^2-x)\,\mathrm dA=\int_0^6\int_0^{6-x}(y^2-x)\,\mathrm dy\,\mathrm dx=72$