x^3 = 2 - x @ x = 1
So from 0 to 1, we have the area under x^3. And from 1 to 2, we have area under 2 - x.
Rational root theorem - Wikipedia, the free encyclopedia
In this case, we can easily see that x=1 is the root we're looking for.
The typical process for finding all roots is to first guess one root using the theorem I linked to above, and then use polynomial long division to factor the polynomial.
For example, if one root is x=1, then you divide the original polynomial with x-1.
And no, your process isn't correct. x=2 and x=-1 aren't roots to the posted polynomial.