find the area of the region included between the parabola y2 = x and the line x + y =2

Here the intersection point is found by putting x = 2-y in y² = x
So we get y = 1,-2
The area can be calculated for the region left of red line by :
$2{\int }_0^1\sqrt{x}dx=2(\frac{1}{2\sqrt{x}} ) from 0 to 1 = 1$

And the area right hand side of red line can be calculated by:
As the curve of y + x = 2 is above y² = x, :
$$\begin{gathered} {\int }_1^2\left[\right(2-x) - \sqrt{x}]dx +{\int }_2^4\left[\right(2-x) - \sqrt{x}]dx = [2x - \frac{x^2}{2} - \frac{1}{2\sqrt{x}}] from 1 to 2 + 2 to 4 \\ = 0.64 + 1.25 = 1.89 \end{gathered}$$
So the total area = 1.89 + 1 = 2.89