Volumes of Revolution (DP IB Maths: AI HL)

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Paul

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Paul

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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 y equals f left parenthesis x right parenthesis
    (and other boundary equations) is rotated 2 pi radians left parenthesis 360 degree right parenthesis around the x-axis
  • The volume of revolution is the volume of this solid

2An illustration showing how a volume of revolution about the x-axis is formed from an area under a curve

  • 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

 V equals pi integral subscript a superscript b y squared space d x

    • This is given in the formula booklet
    • y is a function of x
    • x equals a and x equals b are the equations of the (vertical) lines bounding the area
      • If x equals a and x equals b are not stated in a question, the boundaries could involve the y-axis (x equals 0) and/or a root of y equals f left parenthesis x right parenthesis
      • 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 ofspace y equals f left parenthesis x right parenthesis

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

STEP 2
Use a GDC and the formula to evaluate the integral
Thus find the volume of revolution

Examiner Tip

  • Functions involved can be quite complicated so type them into your GDC carefully
  • Whether a diagram is given or not, using your GDC to plot the curve, limits, etc (where possible) can help you to visualise and make progress with problems

Worked example

Find the volume of the solid of revolution formed by rotating the region bounded by the graph of y equals square root of 3 x squared plus 2 end root, the coordinate axes and the line x equals 3 by 2 pi radians around the x-axis.  Give your answer as an exact multiple of pi.

5-4-4-ib-hl-ai-aa-extraaa-ai-we1-soltn

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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 y equals f left parenthesis x right parenthesis (and other boundary equations) is rotated 2 pi radians left parenthesis 360 degree right parenthesis around the y-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

V equals straight pi integral subscript a superscript b x squared space d y 

    • This is given in the formula booklet
    • x is a function of y
      • the function is usually given in the form y equals f left parenthesis x right parenthesis
      • this will need rearranging into the form x equals g left parenthesis y right parenthesis
    • y equals a and y equals b are the equations of the (horizontal) lines bounding the area
      • If y equals a and y equals b are not stated in the question, the boundaries could involve the x-axis (y equals 0) and/or a root of x equals g left parenthesis y right parenthesis
      • 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 y equals f left parenthesis x right parenthesis
(or x equals g left parenthesis y right parenthesis if already in that form)

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

STEP 2
If needed, rearrange y equals f left parenthesis x right parenthesis into the form x equals g left parenthesis y right parenthesis

STEP 3
Use a GDC and the formula to evaluate the integral
A GDC will likely require the function written with 'x' as the variable (not 'y')
Thus find the volume of revolution 

Examiner Tip

  • Functions involved can be quite complicated so type them into your GDC carefully
  • Whether a diagram is given or not, using your GDC to plot the curve, limits, etc (where possible) can help you to visualise and make progress with problems

Worked example

Find the volume of the solid of revolution formed by rotating the region bounded by the graph of y equals x cubed plus 8 and the coordinate axes by 2 straight pi radians around the y-axis.  Give your answer to three significant figures.

5-4-4-ib-hl-ai-aa-extraaa-ai-we2-vor-soltn

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Paul

Author: Paul

Expertise: Maths

Paul has taught mathematics for 20 years and has been an examiner for Edexcel for over a decade. GCSE, A level, pure, mechanics, statistics, discrete – if it’s in a Maths exam, Paul will know about it. Paul is a passionate fan of clear and colourful notes with fascinating diagrams – one of the many reasons he is excited to be a member of the SME team.