The curve given by  , where  , is rotated about the -axis to form a solid of revolution.

1.  i. Write down the definite integral, in terms of , for the volume of this solid of revolution.  (1 mark)
   

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2. ii. Find the volume of the solid of revolution.  (1 mark)
   

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2.  i. Express the curved surface area of the solid in the form  , where are all positive integers.  (2 marks)
   

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3. ii. Hence or otherwise, find the curved surface area of the solid correct to three decimal places.  (1 mark)
   

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The total surface area of the solid consists of the curved surface area plus the areas of the two circular discs at each end.

The 'efficiency ratio' of a body is defined as its total surface area divided by the enclosed volume.

3. Find the efficiency ratio of the solid of revolution correct to two decimal places.  (2 marks)
   

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4. Another solid of revolution is formed by rotating the curve given by  about the -axis for  , where . This solid has a volume of .
5. Find the efficiency ratio for this solid, giving your answer correct to two decimal places.  (3 marks)
   

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