Suppose i want to calculate the work done by the vector field
$$ F(x,y) =\left(-\frac{y}{x^2 + y^2}, \frac{x}{x^2 + y^2 }\right) $$
when a particle is moving along the ellipse $$ x^2 + \frac{y^2}{4} = 1 $$ from the point (1,0) to (0,-2).
The way i've gone to solve the problem is to enclose the curve by connecting it with the unit circle from (1,0) to (0,-1) and with the line from (0,-1) to (0,-2).
By doing this i've managed to enclose the curve in a way that it is positive orientated. Also since the zone enclosed by the curve doesn't include (0,0) i should be able to use Greens formula.
I get that:
$$ \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y} = 0 $$
so $$ \int\!\!\int_D \frac{\partial Q}{\partial x} - \frac{\partial P}{\partial y}\,dxdy = 0 $$
I then calculate the curve integral from the curves i added to enclose the zone D, and subtract it to arrive to the answer:
$$ W=\int_{\gamma}Pdx+Qdy = \frac{\pi}{2} $$. But the answer should be $$ \frac{3\pi}{2} $$, so im clearly doing something wrong, but i have no idea what.
Any help would be greatly appreciated.