CJC/II/Q4

The diagram shows the region R bounded by the two parabolas 

      Y = x2 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 = x2 into x = (y – 2)2 – 2

y = 2 ±√x+2

x = (x2 – 2)2 – 2

x = x4 – 4x2 + 4 – 2

0 = x4 – 4x2 – x + 2 = (x2 – x – 2)(x2 + x – 1)

X = 2, -1, -1.618, 0.6180

By long division,  Quadratic factor = (x – 2)(x + 1) = x2 – x – 2

 

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

 

 

(a)

(b)

 

 

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