6.2.5 Adding and Subtracting Volumes

Why might I need to add or subtract volumes of revolution? 

• As with the area between a curve and a line or the area between 2 curves, a required volume may be created by two functions
  • In this note we focus on volumes created by rotation around the x-axis but the same principles apply to rotation around the y-axis
  • Make sure you are familiar with the methods in Volumes of Revolution
• The volumes created here can be created from areas that do not have the x-axis as one its boundaries
  • A cylinder is created by rotating a rectangle that borders the x-axis around the x-axis by 360°
  • An annular prism (a cylinder with a whole through it – like a toilet roll) is created by rotating a rectangle that does not have a boundary with the  x‑axis around the x-axis by 360°
• A rectangle would be defined by two vertical and two horizontal lines

• • 
    • Where a, b, c & d are all positive and a < b and c < d 
  • The volume of revolution of this rectangle would be 
    • 

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

• When the area to be rotated around an axis has more than one function (and an axis) 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
    • Whether rotation is around the x-axis or the y-axis 
• Consider the region R, bounded by a curve, a line and the -axis, in the diagram below

• If R is rotated around the -axis the solid of revolution formed will have a ‘hole’ in its centre

• • Think in 2D and area
    • “region under the curve”
      SUBTRACT
      “region under the line” 

• If R is rotated around the -axis the solid of revolution formed will look a little like a spinning top – with a ‘dome top half’ and a ‘cone bottom half’

• • Think in 2D and area
    • “top ‘half’ is the area ‘below’ the curve to the horizontal where the curve and line intersect”
      ADD
      “bottom ‘half’ is area ‘below’ the line to the horizontal where the curve and line interest” 

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 axis that the area will be rotated around
  • Identify the functions  involved in generating the volume       
  • Determine whether these will need to be added or subtracted 

• STEP 2: If rotating around the x-axis, square y for all functions
  • If rotating around the y-axis, rearrange all the y functions into the form  and square
  • In either case do this first without worrying about π or the integration and limits 

• STEP 3: Identify the limits for each volume involved and form the integrals required
  • The limits could come from a graph  

• STEP 4: Evaluate the integral for each function and add or subtract as necessary
  • The answer may be required in exact form
  • If not, round to three significant figures (unless told otherwise)

Worked Example

Exam Tip