5.4.4 Volumes of Revolution

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

• A solid of revolution is formed when an area bounded by a function
  (and other boundary equations) is rotated radians around 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

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

• Use the formula

• • This is given in the formula booklet
  • is a function of
  • and are the equations of the (vertical) lines bounding the area
    • If and are not stated in a question, the boundaries could involve the -axis () and/or a root of
    • Use a GDC to plot the curve, sketch it and highlight the area to help
• Visualising the solid created is helpful
  • Try sketching some functions and their solids of revolution to help

If a diagram is not given, use a GDC to draw the graph of

If not identifiable from the question, use the graph to find the limits and

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

• Very similar to above, this is a solid of revolution which is formed when an area bounded by a function (and other boundary equations) is rotated radians  around the -axis
• The volume of revolution is the volume of this solid

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

• Use the formula

• • This is given in the formula booklet
  •  is a function of 
    • the function is usually given in the form 
    • this will need rearranging into the form
  • and  are the equations of the (horizontal) lines bounding the area
    • If  and  are not stated in the question, the boundaries could involve the -axis () and/or a root of
    • Use a GDC to plot the curve, sketch it and highlight the area to help
• Visualising the solid created is helpful
  • Try sketching some functions and their solids of revolution to help

If a diagram is not given, use a GDC to draw the graph of 

(or  if already in that form)

If not identifiable from the question use the graph to find the limits  and 

If needed, rearrange into the form 

A GDC will likely require the function written with '' as the variable (not '')