A quick question about limits on a line integral involving vector fields.
- Evaluate the line integral $\int_CF\cdot\mathrm ds$ where $F(x,y)=(e^x\sin y+3y,e^x\cos y+2x-2y)$ and $C$ is the ellipse $4x^2+y^2=4$ choosing the counterclockwise direction. (2 points)
I know that the parametrization of this curve is the following
$\begin{align} r(t) &= [\cos(t), 2 \sin(t)]\\ r'(t) &= [-\sin(t), 2 \cos(t)] \end{align}$
and we have our $F(r(t)) = F(x(t), y(t))$ $F(r(t)) = e^{\cos(t)}\sin(2\sin(t))+6\sin(t), e^{\cos(t)}\cos(2\sin(t)) +2\cos(t)-4\sin(t)$
and so by brute force we have the formula for the line integral $ \int_?^? F(r(t)) \cdot r'(t) \,\textrm{d}t $
What would my limits be in this case? A wild guess would be 0 to $2\pi$