Volumes of Revolution (Edexcel IGCSE Further Pure Maths)

Revision Note

Roger B

Written by: Roger B

Reviewed by: Dan Finlay

Volumes of Revolution About the x-axis

What is a volume of revolution about the x-axis?

  • A solid of revolution is formed

    • when an area bounded by a function y equals straight f left parenthesis x right parenthesis and the lines x equals a and x equals b

    • is rotated 2 pi radians left parenthesis 360 degree right parenthesis about the x-axis

  • The volume of revolution is the volume of this solid

Graph rotating around x-axis to form a volume of revolution
  • Be careful – the ’front’ and ‘back’ of this solid are flat

    • they were created from straight (vertical) lines

    • 3D sketches can be misleading!

What is the formula for a volume of revolution about the x-axis?

The volume of revolution of a solid rotated 2 pi radians (360 degree) about the x-axis between x equals a and x equals b is given by:

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

  • This is not given on the exam formula sheet, so you need to remember it

    • Note that pi y squared is the area of the circular cross-section of the solid at any value of x

    • That might help you remember the form of the volume integral

  • y is a function of bold italic x

    • i.e.  y equals straight f open parentheses x close parentheses

  • bold italic x bold equals bold italic a and bold italic x bold equals bold italic b are the equations of the (vertical) lines bounding the area

    • a less than b  (a is the 'left boundary' and b is the 'right boundary')

    • x equals a and x equals b may be given in the question

    • one boundary may be the bold italic y-axis (x equals 0)

    • the bold italic x-axis intercepts of y equals straight f left parenthesis x right parenthesis may also be boundaries

How do I calculate the volume of revolution about the x-axis?

  • STEP 1
    Identify the limits a and b

    • These may be given in the question

      • or be indicated on a graph in the question

    • Sketching the graph of y equals straight f left parenthesis x right parenthesis can help if the graph is not provided

  • STEP 2
    Square the function  y equals straight f open parentheses x close parentheses

    • e.g.  y equals x squared plus 1 space space rightwards double arrow space space y squared equals open parentheses x squared plus 1 close parentheses squared equals x to the power of 4 plus 2 x squared plus 1

    • or  y equals square root of 4 minus x end root space space rightwards double arrow space space y squared equals open parentheses square root of 4 minus x end root close parentheses squared equals 4 minus x

  • STEP 3
    Evaluate the integral in the volume formula V equals pi integral subscript a superscript b y squared space d x

    • An answer may be required in exact form

      • i.e. as a multiple of pi

Examiner Tips and Tricks

  • Don't panic if  y equals straight f open parentheses x close parentheses  involves a square root

    • The square root will disappear when you find y squared

  • Don't forget to bring pi back in after working out the integral

    • In my experience that is a very common student error

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 about the x-axis.  Give your answer as an exact value.

Start by finding the values of a and b for the formula

'Bounded by the coordinate axes' tells us that the y-axis (x equals 0) is one boundary
The question tells us that x equals 3 is the other one
So  a equals 0  and  b equals 3

If in doubt, drawing a sketch can help

Sketch of function for volume of revolution


Now square the function y equals straight f open parentheses x close parentheses

y squared equals open parentheses square root of 3 x squared plus 2 end root close parentheses squared equals 3 x squared plus 2


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

V equals pi integral subscript 0 superscript 3 open parentheses 3 x squared plus 2 close parentheses space straight d x


Work out the definite integral

table row cell integral subscript 0 superscript 3 open parentheses 3 x squared plus 2 close parentheses space straight d x end cell equals cell open square brackets 3 open parentheses fraction numerator x to the power of 2 plus 1 end exponent over denominator 2 plus 1 end fraction close parentheses plus 2 x close square brackets subscript 0 superscript 3 end cell row blank equals cell open square brackets x cubed plus 2 x close square brackets subscript 0 superscript 3 end cell row blank equals cell open parentheses open parentheses 3 close parentheses cubed plus 2 open parentheses 3 close parentheses close parentheses minus open parentheses open parentheses 0 close parentheses cubed plus 2 open parentheses 0 close parentheses close parentheses end cell row blank equals cell 33 minus 0 end cell row blank equals 33 end table

Put that value back into the volume formula

Don't forget to include the pi !

V equals 33 pi


The question asks for an exact value answer, so leave the answer in terms of pi


Volume bold equals bold 33 bold italic pi bold space bold units to the power of bold 3

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Roger B

Author: Roger B

Expertise: Maths

Roger's teaching experience stretches all the way back to 1992, and in that time he has taught students at all levels between Year 7 and university undergraduate. Having conducted and published postgraduate research into the mathematical theory behind quantum computing, he is more than confident in dealing with mathematics at any level the exam boards might throw at you.

Dan Finlay

Author: Dan Finlay

Expertise: Maths Lead

Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.