Volumes of Revolution (CIE A Level Maths: Pure 1)

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Volumes of revolution around the x-axis

What is a volume of revolution around the x-axis? 

  • A solid of revolution is formed when an area bounded by a function space y equals straight f left parenthesis x right parenthesis (and other boundary equations) is rotated 360° around the x-axis
  • A volume of revolution is the volume of this solid formed

2dk4S6Oy_6-2-4-cie-fig1-vol-of-rev

Example of a solid of revolution that is formed by rotating the area bounded by the function y equals f left parenthesis x right parenthesis , the lines x equals a  and  x equals b and the x-axis 360 degree about the x-axis

How do I find the volume of revolution around the x-axis? 

  • To find the volume of revolution created when the area bounded by the function space y equals straight f left parenthesis x right parenthesis, the lines x equals a and x equals b, and the x-axis is rotated 360° about the x-axis use the formula

 V equals pi integral subscript a superscript b y squared d x

  • The formula may look complicated or confusing at first due to the y and dx
    • remember that y is a function of x
    • once the expression for y is substituted in, everything will be in terms of x
  • π is a constant so you may see this written either inside or outside the integral
  • This is not given in the formulae booklet
    • The formulae booklet does list the volume formulae for some common 3D solids – it may be possible to use these depending on what information about the solid is available

How do I solve problems involving volumes of revolution around the x-axis? 

  • Visualising the solid created is helpful
    • Try sketching some functions and their solids of revolution to help
  •  STEP 1 Square y            
    • Do this first without worrying about π or the integration and limits
  • STEP 2 Identify the limits a and b (which could come from a graph)
  • STEP 3 Use the formula by evaluating the integral and multiplying by π
    • The answer may be required in exact form (leave in terms of π)
      • If not, round to three significant figures (unless told otherwise)
  • Trickier questions may give you the volume and ask for the value of an unknown constant elsewhere in the problem

Worked example

6-2-4-cie-fig2-we-solution

Examiner Tip

  • To help remember the formula note that it is only y squared - volume is 3D so you may have expected a cubic expression
    • If rotating a single point around the x-axis a circle of radius would be formed
      • The area of that circle would then be pi y squared
      • Integration then adds up the areas of all circles between a and b creating the third dimension and volume
        (In 2D, integration creates area by adding up lots of 1D lines)

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Volumes of revolution around the y-axis

What is a volume of revolution around the y-axis? 

  • A solid of revolution is formed when an area bounded by a function space y equals straight f left parenthesis x right parenthesis (and other boundary equations) is rotated 360° around the y-axis
  • A volume of revolution is the volume of this solid formed

6-2-4-cie-fig3-vol-of-rev-y-axis

Example of a solid of revolution that is formed by rotating the area bounded by the function y equals f left parenthesis x right parenthesis , the lines y equals c and  y equals dand the x-axis 360 degree about the y-axis

How do I find the volume of revolution around the y-axis? 

  • To find the volume of revolution created when the area bounded by the function space y equals straight f left parenthesis x right parenthesis, the lines y equals c and y equals d, and the y-axis is rotated 360° about the y-axis use the formula

 begin mathsize 22px style V equals pi integral subscript c superscript d x squared d y end style

  • Note that although the function may be given in the form space y equals straight f left parenthesis x right parenthesis it will first need rewriting in the form x equals straight g left parenthesis y right parenthesis 
  • This is not given in the formulae booklet

How do I solve problems involving volumes of revolution around the y-axis? 

  • Visualising the solid created is helpful
    • Try sketching some functions and their solids of revolution to help 

  • STEP 1 Rearrange space y equals straight f left parenthesis x right parenthesis into the form x equals straight g left parenthesis y right parenthesis (if necessary)
    • This is finding the inverse function straight f to the power of negative 1 end exponent left parenthesis x right parenthesis
  • STEP 2 Square x
    • Do this first without worrying about π or the integration and limits
  • STEP 3 Identify the limits c and d (which could come from a graph)
  • STEP 4 Use the formula by evaluating the integral and multiplying by π
    • The answer may be required in exact form (leave in terms of π)
      • If not, round to three significant figures (unless told otherwise) 

  • Trickier questions may give you the volume and ask for the value of an unknown constant elsewhere in the problem

Worked example

6-2-4-cie-fig4-we-solution-part-1

6-2-4-cie-fig4-we-solution-part-2

Examiner Tip

  • Double check questions to ensure you are clear about which axis the rotation is around
  • Separating the rearranging of space y equals straight f left parenthesis x right parenthesisinto x equals straight g left parenthesis y right parenthesis and the squaring of x is important for maintaining accuracy
    • In some cases it can seem as though x has been squared twice
      • in the worked example above, x has been squared twice
      • but it needed to be – once as part of the rearranging, once as part of the volume formula

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.