I am solving a series of problems that begins with, suppose curl $\vec{F}=\langle 5,4y,-2z\rangle$ and $C$ a circle of radius .005 centered at (2,4,5) in the plane $x+y+z=11$. The first part of the assignment is to find $$curl\vec{F}\cdot \langle 1,1,1\rangle$$ at the given point which I easily calculate as 5+16-10=11.
For the next part of the problem it asks me to use that answer to approximate $\int_C \vec{F}\cdot d\vec{r}$. I've never seen a question like this but I'm guessing that since I found the dot with the curl, it has to do with Stoke's Theorem, that $\int_C \vec{F}\cdot d\vec{r}=\iint_S(\nabla\times\vec{F})\cdot d\vec{S}$. My thought is that, since $\langle 1,1,1\rangle$ is normal to the surface, it is a decent approximation of the flow through the surface so that if we dot it with the curl and multiply by the area of the surface we get a good approximation.
But like I said, I took Calc III years ago and never saw a question about approximating anything, so this is a pretty foreign question to me. Please let me know if I've gone wrong anywhere.
[Edit: To be clear, my guess for the approximation is $11\pi(0.005^2)$.