Volumes of Revolution (Edexcel IGCSE Further Pure Maths): Revision Note

Volumes of Revolution About the x-axis

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

• A solid of revolution is formed

  • when an area bounded by a function and the lines and

  • is rotated radians about the -axis

• The volume of revolution is the volume of this solid

• Be careful – the ’front’ and ‘back’ of this solid are flat

  • they were created from straight (vertical) lines

  • 3D sketches can be misleading!

What is the formula for a volume of revolution about the x-axis?

The volume of revolution of a solid rotated  radians () about the -axis between  and  is given by:

• This is not given on the exam formula sheet, so you need to remember it

  • Note that  is the area of the circular cross-section of the solid at any value of

  • That might help you remember the form of the volume integral

• is a function of

  • i.e.  

• and are the equations of the (vertical) lines bounding the area

  •   ( is the 'left boundary' and is the 'right boundary')

  • and may be given in the question

  • one boundary may be the -axis ()

  • the -axis intercepts of may also be boundaries

How do I calculate the volume of revolution about the x-axis?

• STEP 1
  Identify the limits and

  • These may be given in the question

    • or be indicated on a graph in the question

  • Sketching the graph of can help if the graph is not provided

• STEP 2
  Square the function 

  • e.g. 

  • or  

• STEP 3
  Evaluate the integral in the volume formula 

  • An answer may be required in exact form

    • i.e. as a multiple of