You have the right value for the integral you show. If you are calculating area, then you shouldn't get a negative value, so your integral is set up wrong in that case.
If the problem was "Find the area between [itex]x= y^2- 4y[/itex] and [itex]x= 2y- y^2[/itex] for y between 0 and 3" you should note that [itex]2y- y^2> y^2- 4y[/itex] for y between 0 and 3 so you have the subtraction backwards.