[Math] find the area of the shaded region $x=y^2-1, y=1, x=\sqrt y$

Question

find the area of the shaded region $$x=y^2-1, y=1, x=\sqrt y$$
The shaded region contains the point $(0,1)$

$$y=\sqrt{x+1}, y=x^2, y=1$$

$$\int _{-1}^{1}\left(\sqrt{x+1}-1-x^2\right)dx$$

I'm getting something different than the book

Answer 1

Hint:

$$\int_0^1 \sqrt{y}-(y^2-1) \, dy$$