The regions can be visualized as follows:
RegionPlot[{y > x^2, y < 8 - x^2, y > x + 2, y > 1}, {x, -5, 5}, {y, -5, 15}]
Now you can rewrite it and choose the >
and <
symbols correctly:
RegionPlot[y > x^2 \[And] y < 8 - x^2 \[And] y > x + 2 \[And] y > 1, {x, -5, 5}, {y, -5, 15}]
Once you know that go ahead and integrate the area:
Integrate[Boole[y > x^2 \[And] y < 8 - x^2 \[And] y > x + 2 \[And] y > 1], {x, -5, 5}, {y, -5, 15}]