The given curves are: x^{2} = y

Which is an upward parabola with vertex at origin

And line x + y = 2 ⇒ y = 2 – x

x^{2} = 2 – x

⇒ x^{2} + x – 2 = 0

⇒ (x + 2)(x – 1) = 0

⇒ x = -2 and x = 1

Now, y = 2-(-2) = 4

and y = 2 – 1 ⇒ y = 1

⇒ y = 4 and y = 1

Thus, the points of intersection are (-2, 4) and (1, 1)

Required area of shaded region