Compute the area between y=x² and y=8-x² and y=x+2

Questions
1

I want to compute this area with a condition : y>=1. Is it possible?

2

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}]
3

Thank you very much!

4

Note that the y>1 doesn’t do anything…