Yes, if you continue your reasoning, you may be able to get the correct answer.
But it could have been more straightforward (and what follows can be seen on your graph)
If you consider (which is correct), then
So the boundaries for y are :
, which is x, since
The the integral is :
And you can see on your graph, that if you care only of the right parts of R1 and R2, y is always > x, but always below 1/4 - x².
Does it look clear to you ?