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

Parametric volumes of revolution

What is a parametric volume of revolution? 

• Solids of revolution are formed by rotating functions about the x-axis

• Here though, rather than given y in terms of x, both x and y are given in terms of a parameter, t

  • $x=\mathrm{f}\left(t\right)$

  • $y=\mathrm{g}\left(t\right)$ 

  • 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 parametric volumes of revolution?

• The aim is to replace everything in the ‘original’ integral so that it is in terms of t

• For the ‘original’ integral $V=\pi {\int }_{x_1}^{x_2}{y}^{2}dx$ and parametric equations given in the form $x=\mathrm{f}\left(t\right)$ and $y=\mathrm{g}\left(t\right)$ use the following process

• STEP 1: Find dx in terms of t and dt

  • $dx=\mathrm{f}\text{'}\left(t\right)dt$

• STEP 2: If necessary, change the limits from x values to t values using

  • $x_1=\mathrm{f}\left(t_1\right)$

  • $x_2=\mathrm{f}\left(t_2\right)$

• STEP 3: Square y

  • $y^2={\left[\mathrm{g}\right(t\left)\right]}^2$

  • 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

$V=\pi {\int }_{t_1}^{t_2}(\mathrm{g}{\left(t\right))}^2\mathrm{f}\text{'}(t)dt$