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please give me the easiest and most simplest method and solution with specific steps by steps working and with the FULL COMPLETE ANSWER

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.



The diagram shows the region R bounded by the two parabolas y = x^2 and x = (y - 2)^2 - 2 and the y-axis.

(i) Find the coordinates of the points indicated A and B in the diagram.
(ii) Find the area of the region R.
(iii) Find the volume of solid generated when R is rotated 2π radians about the y-axis.

Solve a problem involving finding the area and volume of a region bounded by two parabolas y = x^2 and x = (y - 2)^2 - 2, including coordinates of intersection points. Detailed steps include integration for area and volume calculations. Suitable for advanced high school math students (Grades 11-12).

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