Question

Use the Green's Theorem to calculate the work done by the field
[ F (x, y) = -3y^5 i + 5y^2x^3 j ] to move a particle along the circumference
[C: x^2 + y^2 = 9] starting from the point (2;0) and arriving at the point (-2,0).

Answer 1

Let $R$ be the region bounded by $C$. By Green's theorem,

$\displaystyle\int_C\mathbf F\cdot\mathrm d\,\mathbf r=\iint_R\left(\dfrac{\partial Q}{\partial x}-\dfrac{\partial P}{\partial y}\right)\,\mathrm dA$

where $\mathbf F(x,y)=P(x,y)\,\mathbf i+Q(x,y)\,\mathbf j$. Expressing the area in polar coordinates, you have

$\dfrac{\partial Q}{\partial x}=15x^2y^2=15r^4\sin^2\theta\cos^2\theta$
$\dfrac{\partial P}{\partial y}=-15y^4=-15r^4\sin^4\theta$

$\displaystyle\int_{\theta=0}^{\theta=\pi}\int_{r=0}^{r=3}\left(15r^3\sin^2\theta\cos^2\theta+15r^4\sin^4\theta\right)r\,\mathrm dr\,\mathrm d\theta$
$=\displaystyle15\left(\int_0^\pi\sin^2\theta\,\mathrm d\theta\right)\left(\int_0^3r^5\,\mathrm dr\right)$
$=\dfrac{3645\pi}4$