The diagram shows the region R bounded by the two parabolas

Y = x^{2} and x = (y – 2)^{2} – 2 and the y – axis. Find the points indicated A and B in the diagram.

(a)Find the area of the region R

(b)Find the volume formed when R is rotated 2π radian about the y – axis

Solutions: A(-1 , 1), B(0 , 2 – 2√2)

(a) area = 0.448

(b) volume = 1.10

Sub Y = x^{2} into x = (y – 2)^{2} – 2

y = 2 ±√x+2

x = (x^{2} – 2)^{2} – 2

x = x^{4} – 4x^{2} + 4 – 2

0 = x^{4} – 4x^{2} – x + 2 = (x^{2} – x – 2)(x^{2} + x – 1)

X = 2, -1, -1.618, 0.6180

By long division, Quadratic factor = (x – 2)(x + 1) = x^{2} – x – 2

A(-1 , 1) B( (-1 + √5)/2 , [(-1 + √5)/2]^{2})

(a)

(b)