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Calculus, SPEC2 2022 VCAA 1

Consider the family of functions with rule  , where .

  1. Write down the equations of the two asymptotes of the graph of when .   (2 marks)

  2. Sketch the graph of    for    on the set of axes below. Clearly label any turning points with their coordinates and label any asymptotes with their equations.   (3 marks)
     

  1.  i. Find, in terms of , the equations of the asymptotes of the graph of  .   (1 mark)

  2. ii. Find the distance between the two turning points of the graph of  in terms of .   (2 marks)

  3. Now consider the functions and , where    and  .
  4. The region bounded by the curves of and is rotated about the -axis.
    1. Write down the definite integral that can be used to find the volume of the resulting solid.   (2 marks)

    2. Hence, find the volume of this solid. Give your answer correct to two decimal places.   (1 mark)

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♦♦ Mean mark (d)(ii) 37%.

Calculus, SPEC1 2022 VCAA 10

Let .

  1. Sketch the graph of for on the set of axes below. Label any asymptotes with their equations and label any turning points and the endpoints with their coordinates.   (3 marks)
      

      
  2. The graph of  for is rotated about the -axis to form a solid of revolution.
    Find the volume of this solid. Give your answer in the form , where , , .   (3 marks)
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b.  

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♦ Mean mark (a) 49%.
b.   
   
   
   
   
   
   
   
   
   

♦ Mean mark (b) 55%.

Calculus, SPEC2 2023 VCAA 3

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)

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

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

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

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.

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

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

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Calculus, SPEC1 2023 VCAA 7

The curve defined by the parametric equations

, where 

is rotated about the -axis to form an open hollow surface of revolution.

Find the surface area of the surface of revolution.

Give your answer in the form , where and .   (4 marks)

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Calculus, SPEC1 2021 VCAA 4

  1. The shaded region in the diagram below is bounded by the graph of    and the -axis between the first two non-negative -intercepts of the curve, that is interval  .  The shaded region is rotated about the -axis to form a solid of revolution.
     
           
     
    Find the volume, of the solid formed.   (3 marks)

  2. Now consider the function  , where is a positive real constant. The region bounded by the graph of the function and the -axis between the first two non-negative -intercepts of the graph is rotated about the -axis to form a solid of revolution.
  3. Find the volume of this solid in term of .   (1 mark)

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♦♦ Mean mark part (b) 30%.

 

b.   

Calculus, SPEC2 2017 VCAA 1

Let  , where is the maximal domain of  .

  1.   i. Find the equations of any asymptotes of the graph of  .   (1 mark)

  2.  ii. Find    and state the coordinates of any stationary points of the graph of  , correct to two decimal places.  (2 marks)

  3. iii. Find the coordinates of any points of inflection of the graph of  , correct to two decimal places.  (2 marks)

  4. Sketch the graph of    from    and    on the axes provided below, marking all stationary points, points of inflection and intercepts with axes, labelling them with their coordinates. Show any asymptotes and label them with their equations.  (3 marks)

     

     

  5. The region , bounded by the graph of  , the -axis and the line  , is rotated about the -axis to form a solid of revolution. The line  , where  , divides the region into two regions such that, when the two regions are rotated about the -axis, they generate solids of equal volume.
  6. i.  Write down an equation involving definite integrals that can be used to determine .  (2 marks)

  7. ii. Hence, find the value of , correct to two decimal places.  (1 mark)

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  2. ii.
  3. iii.
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♦♦ Mean mark 36%.


 

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Calculus, SPEC1 2017 VCAA 10

  1.  Show that  , where  .   (1 mark)

  2.  State the maximum domain and the range of  .   (2 marks)

  3.  Find the volume of the solid of revolution generated when the region bounded by the graph of  , and the lines    and  , is rotated about the -axis.   (4 marks)

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♦ Mean mark part (c) 35%.


 

 
 

Calculus, SPEC1-NHT 2018 VCAA 9

  1.    i. Given that  , show that  .   (2 marks)

  2.  ii. Show that  .  (1 mark)

  3. iii. Hence, show that  , given that  , where  .   (1 mark)

  4.  Find the gradient of the tangent to the curve    at  .   (2 marks)

  5.  A solid of revolution is formed by rotating the region between the graph of  , the horizontal axis, and the lines    and    about the horizontal axis.
  6. Find the volume of the solid of revolution.   (3 marks)

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Calculus, SPEC1 2018 VCAA 9

A curve is specified parametrically by  .

  1.  Show that the cartesian equation of the curve is  .   (2 marks)

  2.  Find the -coordinates of the points of intersection of the curve    and the line  .   (1 mark)

  3.  Find the volume of the solid of revolution formed when the region bounded by the curve and the line is rotated about the -axis.   (2 marks)

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b.   
 
 
 
 
 

 

♦♦ Mean mark 30%.

 

c.