What is the value of the line integral?

[brainslush]

Homework Statement

Use Green’s theorem to find the integral
$\oint_{\gamma} \frac{-y}{x^2+y^2}dx+\frac{x}{x^2+y^2}dy$
along two different curves γ: first where γ is the simple closed curve which goes along x = −y2 + 4 and x = 2, and second where γ is the square with vertices (−1, 0), (1, 0), (0, 1), (0, −1).

Homework Equations

The Attempt at a Solution

I'm bit confused b/c
$d(\frac{-y}{x^2+y^2})/dy = \frac{y^2-x^2}{(x^2+y^2)^2}$
$d(\frac{x}{x^2+y^2})/dx = \frac{y^2-x^2}{(x^2+y^2)^2}$

Then by Green's theorem one gets

$\int_{A}\int (d(\frac{x}{x^2+y^2})/dx-d(\frac{-y}{x^2+y^2})/dy) dx dy = \int_{A}\int 0 dx dy = 0$

What am I missing?

 

[Danny B]

You are not missing anything. The answer IS 0.

Here you have a line integral of a conservative field (you can tell it's conservative from the equality of the partial derivatives).
A line integral of a conservative field will always be zero for any path that begins and ends at the same point, i.e any closed curve.