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 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

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 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

STEP 1

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 

STEP 2

If needed, rearrange into the form 

STEP 3

Use a GDC and the formula to evaluate the integral

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

Thus find the volume of revolution