Parametric Volumes of Revolution (Edexcel IAL Maths)

Spec point — 6.1

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

  • $x=f(t)$
  • $y=g(t)$
  • 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 $V=π \int_{x _{1}}^{x _{2}} y² d x$ and parametric equations given in the form $x=f(t)$ and $y=g(t)$ use the following process

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

  • $dx=f'(t)dt$

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

  • $x_{1}=f(t_{1})$
  • $x_{2}=f(t_{2})$

• STEP 3: Square y

  • $y²= \text{stretchy} [ \mathrm{g} ( t ) \text{stretchy} ] ²$
  • 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

$V= \pi \int_{t _{1}}^{t _{2}} ( \mathrm{g} ( t ) ) ² \mathrm{f} ' ( t ) d t$

Worked Example

Exam Hint

• 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