Do a quick sketch of the graph or visualise it in your head. In this case the area will be all above the x axis so no absolute value signs are needed.
Then use simultaneous equations to find the limits, in this case from x=-1 to x=1
Then simply integrate [2 - x^2 - (x^2)]
We get the integral equalling [2x - 2/3*x^3] from -1 to 1
Substituting we get [2-2/3 - (-2+2/3)] = 8/3 units squared