The shaded region shown in the diagram is bounded by the curve  , the -axis and the line  .
  

Find the volume of the solid of revolution formed when the shaded region is rotated about the -axis.  (3 marks)

In the diagram, the shaded region is bounded by the parabola  , the -axis and the line  .

Find the volume of the solid formed when the shaded region is rotated about the -axis.  (3 marks)

The graphs of the curves    and    are shown in the diagram.

1. Find the points of intersection of the two curves.  (1 mark)
   

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2. The shaded region between the curves and the -axis is rotated about the -axis. By splitting the shaded region into two parts, or otherwise, find the volume of the solid formed.  (3 marks)
   

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The diagram shows the region enclosed by the parabola  , the  -axis and the line  , where  . This region is rotated about the  -axis to form a solid called a paraboloid. The point    is the intersection of   and  .

The point    has coordinates  .
 

1. Find the exact volume of the paraboloid in terms of  .    (2 marks)
   

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2. A cylinder has radius    and height  .    
   

    

   What is the ratio of the volume of the paraboloid to the volume of the cylinder?   (1 mark)
   

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The region bounded by the  -axis, the  -axis and the parabola    is rotated about the  -axis to form a solid.
 

Find the volume of the solid.   (4 marks)