Parametric Volumes 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

The aim is to replace everything in the ‘original’ integral so that it is in terms of t
For the ‘original’ integral $V = π \int_{x_{1}}^{x_{2}} y^{2} d x$ 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
- $d x = f' (t) d t$
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
- $y^{2} = {[g (t)]}^{2}$
- 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

$V = π \int_{t_{1}}^{t_{2}} (g {(t))}^{2} f' (t) d t$

6-2-4-ial-fig1-we-solution

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

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