Your error is not completing the boundary- the integral around the semi-circle is equal to the integral over the area minus the integral along the x-axis.
But the obvious way to do it is directly- use parametric equations: [itex]x= cos(\theta)[/itex], '[itex]y= sin(\theta)[/itex], [itex]dx= -sin(\theta)d\theta[/itex], [itex]dy= cos(\theta)d\theta[/itex], [itex]\theta[/itex] from [itex]\pi[/itex] to [itex]2\pi[/itex] (assuming you are integrating from (-1, 0) to (1, 0), you don't mention the orientation).