Question

Use Green's Theorem to find the work done by the force F(x, y) = x(x + y) i + xy^2 j in moving a particle from the origin along the x-axis to (4, 0), then along the line segment to (0, 4), and then back to the origin along the y-axis.

Answer 1

By Green's theorem, the work done by $\vec F$ over the path (call it $C$) is

$W=\displaystyle\int_C(x(x+y)\,\vec\imath+xy^2\,\vec\jmath)\cdot\mathrm d\vec r=\int_Cx(x+y)\,\mathrm dx+xy^2\,\mathrm dy$

$=\displaystyle\iint_D\frac{\partial(xy^2)}{\partial x}-\frac{\partial(x(x+y))}{\partial y}\,\mathrm dA$

(where $D$ denotes the region bordered by $C$)

$=\displaystyle\int_0^4\int_0^{4-x}y^2-x\,\mathrm dy\,\mathrm dx=\boxed{\frac{32}3}$