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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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a.  
       

b.  

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a.      
       


♦ Mean mark (a) 49%.
b.   
   
   
   
   
   
   
   
   
   

♦ Mean mark (b) 55%.

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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  1. ³

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a.   
   
   
   
   
    ³

♦♦ Mean mark part (b) 30%.

 

b.   

Calculus, SPEC2-NHT 2017 VCAA 1

  1. i.  Use an appropriate double angle formula with    to deduce a quadratic equation of the form  , where and are real values.   (2 marks)

  2. ii. Hence show that  .   (1 mark)

Consider  .

  1. Sketch the graph of on the axes below, labelling the end points with their coordinates.   (3 marks)


 

  1. The region between the graph of    and the -axis is rotated about the -axis to form a solid of revolution.
  2. i.  Write down a definite integral in terms of    that gives the volume of the solid formed.   (2 marks)

  3. ii. Find the volume of the solid, correct to the nearest integer.   (1 mark)

  1. A fish pond that has a shape approximately like that of the solid of revolution in part c. is being filled with water. When the depth is metres, the volume, , of water in the pond is given by

     

     

    If water is flowing into the pond at a rate of 0.03 m³ per minute, find the rate at which the depth is increasing when the depth is 0.6 m. Give your answer in metres per minute, correct to three decimal places.   (3 marks)

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  1. i. 
  2. ii.
  4. i. 
  5. ii.
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a.i.   

 

 

a.ii.   
 
 
 

 

 

b.   


 

 

c.i.   
 
 
 
 

 

 
   

 

c.ii. 

³

 

d.   
 
 
 
   

  

Calculus, SPEC2-NHT 2018 VCAA 1

Consider the function    with rule  .

  1.  Sketch the graph of    over its maximal domain on the set of axes below. Label the endpoints with their coordinates.   (3 marks)

     


 

  1. A vase is to be modelled by rotating the graph of    about the -axis to form a solid of revolution, where units of measurement are in centimetres.
    1.  Write down a definite integral in terms of that gives the volume of the vase.   (2 marks)

    2.  Find the volume of the vase in cubic centimetres.  (1 mark)

  3. Water is poured into the vase at a rate of 20 cm³ s¯¹.
  4. Find the rate, in centimetres per second, at which the depth of the water is changing when the depth is    cm.  (3 marks)

  5. The vase is placed on a table. A bee climbs from the bottom of the outside of the vase to the top of the vase.
  6. What is the minimum distance the bee will need to travel? Give your answer in centimetres, correct to one decimal place.  (1 mark)

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  2.  i.
  3. ii.

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a.  

 

b.i.  
 
 
 
 

 

 
   

 

b.ii.  

 

c.  ³

 


 

 
 

 

d.   

 

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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a.i.  
 
 
 
 
 

 

a.ii.  
 
 
 

 

a.iii.   

 
 
 
   
   

 

 

b.  
 
   

 

 
 
 

 

c.  
   
   
   
   
   
   
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