Compute the line integral \(\int_{\mathbf{c}} \mathbf{F} \cdot d \mathbf{r}\) for the given vector field and path.

\(\mathbf{F}(x, y)=\left\langle 2 x y, x^{2}+y^{2}ightangle\), the part of the unit circle in the first quadrant oriented counterclockwise