By Green's theorem (all the conditions are met), we have
[tex]\displaystyle \int_C \sqrt y \, dx + \sqrt x \, dy = \iint_D \frac{\partial(\sqrt x)}{\partial x} - \frac{\partial(\sqrt y)}{\partial y} \, dx \, dy[/tex]
where D is the interior of the path C, or the set
[tex]D = \left\{ (x, y) : 0 \le y \le \dfrac{x^2}2 \text{ and } 0 \le x \le 2 \right\}[/tex]
So, the line integral reduces to the double integral,
[tex]\displaystyle \frac12 \int_0^2 \int_0^{\frac{x^2}2} x^{-\frac12} - y^{-\frac12} \, dy \, dx[/tex]
[tex]\displaystyle = \frac12 \int_0^2 x^{-\frac12}\left(\frac{x^2}2\right) - 2\left(\frac{x^2}2\right)^{\frac12} \, dx[/tex]
[tex]\displaystyle = \frac12 \int_0^2 \frac12 x^{\frac32} - \sqrt 2 \, x \, dx[/tex]
[tex]\displaystyle = \frac14 \int_0^2 x^{\frac32} - 2\sqrt 2 \, x \, dx[/tex]
[tex]\displaystyle = \frac14 \left(\frac25\cdot2^{\frac52} - \sqrt2\cdot2^2\right) = \boxed{-\frac{3\sqrt2}5}[/tex]
