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

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

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 $V=\pi {\int }_a^b{y}^{2}dx$

y-axis $V=\pi {\int }_c^d{x}^{2}dy$

• 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)