So every time I use Green's theorem to calculate the area… I get $0$; here is the curve:
$x = 2\cos(t)$ , $y =\sin(2t)$, $0 \le t \le 2\pi$.
The equation I am using is $$A = \frac{1}{2} \int_C xdy – ydx.$$
Perhaps I am using the wrong equation? I basically plug in $x, y , dy$, and $dx$ and then integrate with respect to $t$ from the given parameter. Is this wrong?
Best Answer
No, changing the bounds will not help you here. The area is signed (or "oriented"). An easy way to look at this is to form the cross product $dA = r \times v \, dt$, where $v = dr/dt$. You should be able to see that, for $x > 0$, this cross product is in the positive $z$-direction. For $x < 0$, this cross product is in the negative $z$-direction. The two halves have opposite orientations.
By symmetry, both halves have the same area, but since they're oriented in opposite directions, they cancel out, and you end up with zero.