Volumes of Revolution (Edexcel IGCSE Further Pure Maths): Revision Note

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 $y=\mathrm{f}\left(x\right)$ and the lines $x=a$ and $x=b$

  • is rotated $2\pi$ radians $(360°)$ about the $x$-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!

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

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

• $V=\pi {\int }_a^b{y}^{2}dx$

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

  • Note that $\pi y^2$ 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 $\mathit{x}$

  • i.e.  $y=\mathrm{f}\left(x\right)$

• $\mathit{x}\mathbf{=}\mathit{a}$ and $\mathit{x}\mathbf{=}\mathit{b}$ are the equations of the (vertical) lines bounding the area

  • $a<b$  ($a$ is the 'left boundary' and $b$ is the 'right boundary')

  • $x=a$ and $x=b$ may be given in the question

  • one boundary may be the $\mathit{y}$-axis ($x=0$)

  • the $\mathit{x}$-axis intercepts of $y=\mathrm{f}\left(x\right)$ 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 $y=\mathrm{f}\left(x\right)$ can help if the graph is not provided

• STEP 2
  Square the function  $y=\mathrm{f}\left(x\right)$

  • e.g.  $y=x^2+1\Rightarrow y^2={\left(x^2+1\right)}^2=x^4+2x^2+1$

  • or  $y=\sqrt{4-x}\Rightarrow y^2={\left(\sqrt{4-x}\right)}^2=4-x$

• STEP 3
  Evaluate the integral in the volume formula $V=\pi {\int }_a^b{y}^{2}dx$

  • An answer may be required in exact form

    • i.e. as a multiple of $\pi$