Volumes of Revolution (Edexcel International A Level (IAL) Maths: Pure 4): Revision Note

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 $y=\mathrm{f}\left(x\right)$ (and other boundary equations) is rotated 360° around the x-axis

• A volume of revolution is the volume of this solid formed

Example of a solid of revolution that is formed by rotating the area bounded by the function $y=f\left(x\right)$ , the lines $x=a$ and  $x=b$ and the $x$-axis $360°$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 $y=\mathrm{f}\left(x\right)$, the lines $x=a$ and $x=b$, and the x-axis is rotated 360° about the x-axis use the formula

 $V=\pi {\int }_a^b{y}^{2}dx$

• 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

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