Let's solve each part of this problem step-by-step.

Given:

We have two curves:

1. $y = x^2$
2. $x = (y - 2)^2 - 2$

The region $R$ is bounded by these curves and the $y$-axis.

Solution

Part (i): Find the coordinates of points $A$ and $B$

1. Find the intersection points of the curves $y = x^2$ and $x = (y - 2)^2 - 2$.

   Since $A$ and $B$ lie on both curves, we set $y = x^2$ into $x = (y - 2)^2 - 2$:

   Substitute $y = x^2$: $x = (x^2 - 2)^2 - 2$

2. Simplify the equation:

   Expanding $(x^2 - 2)^2$: $x = (x^2 - 2)(x^2 - 2) = x^4 - 4x^2 + 4$ So we have: $x = x^4 - 4x^2 + 4 - 2$ $x = x^4 - 4x^2 + 2$ Rearrange to get a polynomial equation: $x^4 - 4x^2 - x + 2 = 0$

   Solving this equation will give us the $x$-coordinates of points $A$ and $B$. However, let's look at approximate solutions or other simpler ways.