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Calculus, EXT1 C3 SM-Bank 1

The region enclosed by the semicircle    and the -axis is to be divided into two pieces by the line  , when  .
 


 

The two pieces are rotated about the -axis to form solids of revolution. The value of is chosen so that the volumes of the solids are in the ratio .

Show that satisfies the equation  .  (3 marks)

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(i)   

 

Calculus, EXT1* C3 2016 HSC 15a

The diagram shows two curves  and The curve is the semicircle   The curve has equation  
 

hsc-2016-15a
 

An egg is modelled by rotating the curves about the -axis to form a solid of revolution.

Find the exact value of the volume of the solid of revolution.  (4 marks)

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³

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  ³

 

 

 
 
  ³

 

 
  ³

Calculus, EXT1* C3 2010 HSC 10b

The circle    has radius and centre . The circle meets the positive -axis at . The point is on the interval . A vertical line through meets the circle at . Let  .
  

2010 10b1

  1. The shaded region bounded by the arc and the intervals and is rotated about the -axis. Show that the volume, , formed is given by
  2.   (3 marks)

  3. A container is in the shape of a hemisphere of radius metres. The container is initially horizontal and full of water. The container is then tilted at an angle of to the horizontal so that some water spills out. 

  1. (1) Find so that the depth of water remaining is one half of the original depth.   (1 mark)

  2. (2) What fraction of the original volume is left in the container?   (2 marks)

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  2. (1)
  3. (2)
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i.   

♦♦♦ Mean mark (i) 16%.
MARKER’S COMMENT: A common error was to integrate to instead of (note that is a constant).  

 

 
 
 
 
 

 

ii. (1)

♦♦♦ Part (ii) mean marks 3% and 2% for (ii)(1) and (ii)(2) respectively.

 

 MARKER’S COMMENT: Previous parts of a question should always be at the front and centre of a student’s mind and direct their strategy.
ii. (2)