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

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Modelling with Volumes of Revolution

What is meant by modelling with volumes of revolution? 

  • Many every day objects such as buckets, beakers, vases and lamp shades can be modelled as a solid of revolution
  • This can then be used to find the volume of the solid (volume of revolution) and/or other information about the solid that could be useful before an object is manufactured
  • Modelling with volumes of revolution could involve rotation around the x-axis or y-axis so ensure you are familiar with both

6-2-6-cie-fig1-bowl-and-lamp-1

What modelling assumptions are there with volumes of revolution? 

  • The solids formed are usually the main shape of the body of the object
    • For example, the handles on a vase would not be included
    • The lip on the top edge of a bucket would not be considered
  • A common question or assumption concerns the thickness of a container
    • The thickness is generally ignored as it is relatively small compared to the size of the object
      • thickness will depend on the purpose of the object and the material it is made from
    • Some questions may refer to the solid formed being the ‘inside’ of an object or refer to the ‘internal’ dimensions
    • If the thickness of the material is significant it would involve two related solids of revolution (Adding & Subtracting Volumes)

How do I solve modelling problems with volumes of revolution? 

  • Visualising and sketching the solid formed can help with starting problems
  • Familiarity with applying the volume of revolution formulae for rotations around both the x and y axes

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

y-axis V equals pi integral subscript c superscript d x squared d y

  • The volume of a solid may involve adding or subtracting different volumes of revolution
    • Subtraction would need to be used for solids formed from areas that do not have a boundary with the axis of rotation
  • Questions may go on to ask related questions in context so do take notice of the context
    • A question about a bucket being formed may ask about its capacity
  • This would be measured in litres so there may also be a mix of units that will need conversion (e.g. 1000cm3 = 1 litre)

Worked example

6-2-6-cie-fig2-we-solution-part-1

6-2-6-cie-fig2-we-solution-part-2

Examiner Tip

  • Consider the context of the question to gauge whether your final answer is realistic
    • a vase holding just 0.126 litres of water will not hold many flowers, but the question did state it was a miniature vase
  • For rotation around the y -axis remember to rearrange space y equals straight f left parenthesis x right parenthesis into the form space x equals straight g left parenthesis y right parenthesis

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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.