finding limits of a line integral with vector fields

Question

so just a quick question about limits on a line integral involving vector fields.

Here is the question at hand: http://imgur.com/Qka2N

So what I know and have done. I know that the parametrization of this curve is the following $$r(t) = [\cos(t), 2 \sin(t)]$$ $$r'(t) = [-sin(t), 2 cos(t)]$$

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 $2\pi$

Answer 1

The easiest way to determine the limits of a line integral is simply to look at the function $r(t)$. Since your curve $C$ is an ellipse, your wild guess is right, because $r(t)$ runs over $C$ only once when $r(t)$ runs from $0$ to $2 \pi$. The limits do not depend on the vector field $F$, they must only be such that $r(t)$ runs over $C$ once when $t$ goes over the chosen domain of $r$. What I mean by "once" is that for instance had we chosen $0$ and $4 \pi$ for the limits, you would run over $C$ twice.

Did you just need confirmation or were you actually wondering about something more?

Hope that helps,