Finding circulation and flux for vector fields without given boundaries

[Differentiate1]

Here's my work: http://i.imgur.com/9ik31P5.png

I need confirmation that the vector fields I found for parts 1) and 2) are correct. I also need to figure out how to find the flux in part 2) when the problem doesn't give any boundaries to evaluate. For part 1), the answer can be found by concluding that the vector field F is conservative (zero circulation).

Thanks in advance,

Differentiate1

 

[BruceW]

your answer for part 1) looks good to me. For part 2), I really don't understand the question. It says to imagine integral $I$ as a flux integral for a vector field $F$ but that doesn't make any sense, because integral $I$ is definitely a line integral, as you used it in part 1). The integral $I$ can't be both a line integral and a surface integral, so I don't know what is happening there.

 

[vanhees71]

Well, you can in fact relate a surface integral and a line integral along its closed boundary curve. Just think about the integral theorems of vector calculus :-).

 

[BruceW]

hmm. That's true. But the thing that is being related is only the curve. In the question, there is no curve given. So there is nothing to relate.