(a) The figure on the right shows the ellipse 
and the line x = 1. By using the substitution 
, find the area of the shaded region, giving your answer in an exact form.
(b) Find the exact volume generated when the shaded region is rotated through 2π radians about the y-axis.
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)
The curve C is defined by the equations
(i) Sketch C, showing all axial-intercepts and endpoints clearly.
(ii) Using the fact that C is periodic with period 2π, or otherwise, find the exact area enclosed by C, the line x=-2π, x=2π and the x-axis.
(iii) C1 is the part of the curve C for π≤θ≤2π. The region R is bounded by C1, the axes and the line y=2. State the area of R.
(iv) Find the volume of the solid formed when R is rotated through 2π radians about the y-axis, giving your answer to 1 decimal place.
(v) Find the volume of the solid formed when R is rotated through 2π radians about the x-axis, giving your answer to 2 decimal places.
The curve C has parametric equations
(i) Sketch C, labeling all axial intercepts(s).
(ii) Find the equation of the tangent to the curve when t=1, leaving your answer in terms of e.
(iii) The normal to the curve C at point 
is parallel to the x-axis and intersects the y-axis at point S (0,s), where s<1. Find the exact value of p.
(iv) Find the area of the region bounded by the curve C and the y-axis.
