Volumes of Revolution (DP IB Applications & Interpretation (AI)): Revision Note

Volumes of revolution around x-axis

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 a volume of revolution around the x-axis?

• Use the formula

• 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 for practice

• STEP 1

  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
   

• STEP 2

  Use a GDC and the formula to evaluate the integral

  Thus find the volume of revolution

Volumes of revolution around y-axis

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 a volume of revolution around the y-axis?

• Use the formula

•  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 for practice

  • Visualising the solid created is helpful

    • Try sketching some functions and their solids of revolution to help

• STEP 1

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

  (or  if the function is already given in that form)

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

• STEP 2

  If needed, rearrange into the form 
   

• STEP 3

  Use a GDC and the formula to evaluate the integral

  (Your GDC will likely require you to enter the function written with '' as the variable instead of '')

  Thus find the volume of revolution