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((\partial Q)/(\partial x)-(\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


`(\partial Q)/(\partial x)=15x^2y^2=15r^4\sin^2\theta\cos^2\theta`

`(\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)`

`=\frac{3645\pi}4`