(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 […]

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 […]

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. […]