Volumes of Revolution (Edexcel A Level Further Maths: Core Pure)

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Volumes of revolution around the 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 space y equals straight f left parenthesis x right parenthesis (and other boundary equations) is rotated 360° around the x-axis
  • A volume of revolution is the volume of this solid formed

2dk4S6Oy_6-2-4-cie-fig1-vol-of-rev

Example of a solid of revolution that is formed by rotating the area bounded by the function y equals f left parenthesis x right parenthesis , the lines x equals a  and  x equals b and the x-axis 360 degree about the x-axis

How do I find the volume of revolution around the x-axis? 

  • To find the volume of revolution created when the area bounded by the function space y equals straight f left parenthesis x right parenthesis, the lines x equals a and x equals b, and the x-axis is rotated 360° about the x-axis use the formula

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

  • The formula may look complicated or confusing at first due to the y and dx
    • remember that y is a function of x
    • once the expression for y is substituted in, everything will be in terms of x
  • π is a constant so you may see this written either inside or outside the integral
  • This is not given in the formulae booklet
    • The formulae booklet does list the volume formulae for some common 3D solids – it may be possible to use these depending on what information about the solid is available

Where does the formula for the volume of revolution come from?

  • When you integrate to find the area under a curve you can see the formula by splitting the area into rectangles with small widths
    • The same method works for volumes
  • Split the volume into cylinders with small widths
    • The radius will be the y value
    • The width will be a small interval along the x-axis δx
  • The volume can be approximated by the sum of the volumes of these cylinders

V almost equal to sum pi y squared delta x

  • The limit as δx goes to zero can be found by integration - just like with areas

limit as delta x rightwards arrow 0 of sum pi y squared delta x equals integral subscript a superscript b pi y squared d x

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

  • Visualising the solid created is helpful
    • Try sketching some functions and their solids of revolution to help
  •  STEP 1 Square y            
    • Do this first without worrying about π or the integration and limits
  • STEP 2 Identify the limits a and b (which could come from a graph)
  • STEP 3 Use the formula by evaluating the integral and multiplying by π
    • The answer may be required in exact form (leave in terms of π)
      • If not, round to three significant figures (unless told otherwise)
  • Trickier questions may give you the volume and ask for the value of an unknown constant elsewhere in the problem

Examiner Tip

  • To help remember the formula note that it is only y squared - volume is 3D so you may have expected a cubic expression
    • If rotating a single point around the x-axis a circle of radius would be formed
      • The area of that circle would then be pi y squared
      • Integration then adds up the areas of all circles between a and b creating the third dimension and volume
        (In 2D, integration creates area by adding up lots of 1D lines)

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.

5-1-1-edx-a-fm-we1-soltn

Volumes of revolution around the y-axis

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

  • A solid of revolution is formed when an area bounded by a function space y equals straight f left parenthesis x right parenthesis (and other boundary equations) is rotated 360° around the y-axis
  • A volume of revolution is the volume of this solid formed

6-2-4-cie-fig3-vol-of-rev-y-axis

Example of a solid of revolution that is formed by rotating the area bounded by the function y equals f left parenthesis x right parenthesis , the lines y equals c and  y equals dand the x-axis 360 degree about the y-axis

How do I find the volume of revolution around the y-axis? 

  • To find the volume of revolution created when the area bounded by the function space y equals straight f left parenthesis x right parenthesis, the lines y equals c and y equals d, and the y-axis is rotated 360° about the y-axis use the formula

 begin mathsize 22px style V equals pi integral subscript c superscript d x squared d y end style

  • Note that although the function may be given in the form space y equals straight f left parenthesis x right parenthesis it will first need rewriting in the form x equals straight g left parenthesis y right parenthesis 
  • This is not given in the formulae booklet

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

  • Visualising the solid created is helpful
    • Try sketching some functions and their solids of revolution to help 

  • STEP 1 Rearrange space y equals straight f left parenthesis x right parenthesis into the form x equals straight g left parenthesis y right parenthesis (if necessary)
    • This is finding the inverse function straight f to the power of negative 1 end exponent left parenthesis x right parenthesis
  • STEP 2 Square x
    • Do this first without worrying about π or the integration and limits
  • STEP 3 Identify the limits c and d (which could come from a graph)
  • STEP 4 Use the formula by evaluating the integral and multiplying by π
    • The answer may be required in exact form (leave in terms of π)
      • If not, round to three significant figures (unless told otherwise) 

  • Trickier questions may give you the volume and ask for the value of an unknown constant elsewhere in the problem

Examiner Tip

  • Double check questions to ensure you are clear about which axis the rotation is around
  • Separating the rearranging of space y equals straight f left parenthesis x right parenthesisinto x equals straight g left parenthesis y right parenthesis and the squaring of x is important for maintaining accuracy
    • In some cases it can seem as though x has been squared twice

Worked example

Find the volume of the solid of revolution formed by rotating the region bounded by the graph of y equals arcsin space open parentheses 2 x plus 1 close parentheses and the coordinate axes by 2 straight pi radians around the y-axis.  Give your answer to three significant figures.

5-1-1-edx-a-fm-we2-soltn

Volumes of Revolution using Parametric Equations

What is parametric volumes of revolution? 

  • Solids of revolution are formed by rotating functions about the x-axis or the y-axis
  • Here though, rather than given y in terms of x, both x and y are given in terms of a parameter, t
    • space x equals straight f left parenthesis t right parenthesis
    • space y equals straight g left parenthesis t right parenthesis 
    • Depending on the nature of the functions f and g it may not be convenient or possible to find y in terms of x

How do I find volumes of revolution when x and y are given parametrically?

 

  • The aim is to replace everything in the ‘original’ integral so that it is in terms of t
  • For the ‘original’ integral V equals pi integral subscript x subscript 1 end subscript superscript x subscript 2 end superscript space y squared d x or V equals pi integral subscript y subscript 1 end subscript superscript y subscript 2 end superscript space x squared d y and parametric equations given in the form space x equals straight f left parenthesis t right parenthesis and space y equals straight g left parenthesis t right parenthesis use the following process
  • STEP 1: Find dx or dy in terms of t and dt
    • d x equals straight f apostrophe left parenthesis t right parenthesis d t or d y equals straight g apostrophe left parenthesis t right parenthesis d t
  • STEP 2: If necessary, change the limits from x values or y values to t values using
    • x subscript 1 equals straight f stretchy left parenthesis t subscript 1 stretchy right parenthesis or y subscript 1 equals straight g stretchy left parenthesis t subscript 1 stretchy right parenthesis
    • x subscript 2 equals straight f stretchy left parenthesis t subscript 2 stretchy right parenthesis or y subscript 2 equals straight g stretchy left parenthesis t subscript 2 stretchy right parenthesis
  • STEP 3: Square or x
    • y squared equals stretchy left square bracket straight g left parenthesis t right parenthesis stretchy right square bracket squared or x squared equals stretchy left square bracket straight f left parenthesis t right parenthesis stretchy right square bracket squared
    • Do this separately to avoid confusing when putting the integral together
  • STEP 4: Set up the integral, so everything is now in terms of t, simplify where possible and evaluate the integral to find the volume of revolution

V equals pi integral subscript t subscript 1 end subscript superscript t subscript 2 end superscript space left parenthesis straight g left parenthesis t right parenthesis right parenthesis squared straight f apostrophe left parenthesis t right parenthesis d t (if around x-axis) or V equals pi integral subscript t subscript 1 end subscript superscript t subscript 2 end superscript space left parenthesis straight f left parenthesis t right parenthesis right parenthesis squared straight g apostrophe left parenthesis t right parenthesis d t (if around y-axis)

 

Examiner Tip

  • Avoid the temptation to jump straight to STEP 4
    • There could be a lot to change and simplify in exam style problems
    • Doing each step carefully helps maintain high levels of accuracy

Worked example

The curve is defined parametrically by x equals sec open parentheses t close parentheses and y equals square root of cosec open parentheses t close parentheses end rootis rotated 360° about the x-axis between the values of x equals square root of 2 and x equals 2. Show that the volume of the solid of revolution generated by this rotation is pi open parentheses square root of p minus q close parentheses cubic units where p and q are integers to be found.

BSS1ItEt_5-1-1-edx-a-fm-we3-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.