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Let's break down each part of this question.

The region \( R \) is bounded by:
1. The curve \( y = x^2 \).
2. The curve \( x = (y - 2)^2 - 2 \).
3. The \( y \)-axis.

### (i) Coordinates of Points \( A \) and \( B \)

To find points \( A \) and \( B \), we need to determine the intersections of the curves \( y = x^2 \) and \( x = (y - 2)^2 - 2 \), as well as their intersection with the \( y \)-axis.

1. **Intersection of \( y = x^2 \) and \( x = (y - 2)^2 - 2 \):**
   Substitute \( y = x^2 \) into \( x = (y - 2)^2 - 2 \):
   \[
   x = ((x^2 - 2)^2 - 2
   \]


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) [2 marks] Find the (both x- and y-) coordinates of the points indicated A and B in the diagram. You do not need to show your working. (ii) [2 marks] Find the area of the region R. You may give exact value or two decimal places in your answers. (iii) [2 marks] Find the volume of solid generated when R is rotated 2π radians about the y-axis. Give two decimal places in your answer.

Solve a problem involving the area and volume of a region bounded by two parabolas and the y-axis. This solution includes finding intersection points, calculating area between curves, and determining the volume of rotation about the y-axis. Suitable for high school students.

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