Closed line integral along a curve symmetric about the X and Y axes

From the defintion of a closed Line Integral we know that if the F (vector) is conservative that the integral will = 0, and will always be indipendent of path.

What if, however, we have a closed line integral along a curve symmetric about the X and Y axes?

Sorry, i don't know how to make neat equations like other posters.

For example,

Evaluate the closed Integral of (y^2dx + x^2dy) clockwize about

the curve |x| + |y| = 1

Clearly this has 4 cases in the XY domain.

Case 1: x + y = 1 => y = 1-x => dy = -dx

Case 2: x - y = 1 => y = x-1 => dy = dx

Case 3: -x - y = 1 => y = -x-1 => dy = -dx

Case 4: -x + y = 1 => y = 1 + x => dy = dx

I wont bother going through all the line integrals with you. But I do the line integrals with each case with respect to X.

So the X bounds for:

Case 1: 0-->1

Case 2: 1-->0

Case 3: 0-->-1

Case 4: -1--> 0

Then I compute and add them up and it equals to 0. This promted this question because my intial function is not conservative. So I must ask is a closed line integral along a curve symmetric about the X and Y axes = 0? Or did I compute this incorrectly somewhere down the road?

Cheers