Modelling with Volumes of Revolution (DP IB Analysis & Approaches (AA)): Revision Note

Adding & subtracting volumes

When would volumes of revolution need to be added or subtracted?

• The ‘curve’ boundary of an area may consist of more than one function of

  • For example

    • the ‘curve’ boundary from to is

    • the ‘curve’ boundary from to is

  • So the total volume would be 

• The solid of revolution may have a ‘hole’ in it

  • e.g. a ‘toilet roll’ shape would be the difference of two cylindrical volumes

How do I know whether to add or subtract volumes of revolution?

• When the area to be rotated around the -axis has more than one function defining its boundary it can be trickier to tell whether to add or subtract volumes of revolution

  • It will depend on the nature of the functions and their points of intersection

  • Sketch the graph of the functions (using your GDC where possible) and highlight the area required

How do I solve problems involving adding or subtracting volumes of revolution?

• Visualising the solid created becomes increasingly useful (but also trickier) for shapes generated by separate volumes of revolution

  • Continue trying to sketch the functions and their solids of revolution to help

• STEP 1

  Identify the functions involved in generating the volume

  Determine whether the separate volumes will need to be added or subtracted

  Identify the limits for each volume involved

  • Sketching the graphs of and , or using a GDC to do so, is helpful, especially when the limits are not given directly in the question 

• STEP 2

  Square  for all functions 

  • This step is not essential if a GDC can be used to calculate integrals and an exact answer is not required.

• STEP 3

  Using the appropriate volume of revolution formula for each part, evaluate the definite integral and add or subtract as necessary

  • The answer may be required in exact form

Modelling with volumes of revolution

What is meant by modelling volumes of revolution?

• Many everyday objects such as buckets, beakers, vases and lampshades can be modelled as solids of revolution

• The volume of revolution of the solid can then be calculated

• An object that would usually stand upright may be modelled horizontally so its volume of revolution can more easily be found

What modelling assumptions are there with volumes of revolution?

• The solids formed are usually the main part of the body of the object

  • For example, the handle on a bucket would not be included

• For something like a bucket or drinking glass, the thickness of the walls of the object is usually assumed to be negligible relative to the size of the object

  • Thickness will depend on the purpose of the object and the material it is made from

How do I solve modelling problems with volumes of revolution?

• Visualising and sketching the solid formed can help with starting problems

• Be familiar with applying the volume of revolution formulas

  • around the x-axis: 

  • around the y-axis: 

• The volume of revolution may involve adding or subtracting partial volumes

• Questions may ask related questions in context

  • E.g. a question about a bucket may ask about its capacity

    • this would be measured in litres

    • so a conversion of units may be required

    • (1000 cm³ = 1 litre)