Parametric Volumes of Revolution

What is 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

How do I find volumes of revolution when x and y are given parametrically?

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

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

Worked example

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

Examiner Tip

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)

Revision Note

What is parametric volumes of revolution?

How do I find volumes of revolution when x and y are given parametrically?

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1. Proof

1.1 Proof by Contradiction

1.1.1 Proof by Contradiction

2. Algebra & Functions

2.1 Partial Fractions

2.1.1 Linear Denominators

2.1.2 Squared Linear Denominators

3. Sequences & Series

3.1 General Binomial Expansion

3.1.1 General Binomial Expansion

3.1.2 General Binomial Expansion - Subtleties

3.1.3 General Binomial Expansion - Multiple

3.1.4 Approximating Values

4. Differentiation

4.1 Further Applications of Differentiation

4.1.1 Connected Rates of Change

4.2 Implicit Differentiation

4.2.1 Implicit Differentiation

5. Integration

5.1 Further Integration

5.1.1 Substitution (Reverse Chain Rule)

5.1.2 Harder Substitution

5.1.3 Integration by Parts

5.1.4 Integration using Partial Fractions

5.1.5 Volumes of Revolution

5.1.6 Adding and Subtracting Volumes

5.1.7 Modelling with Volumes of Revolution

5.1.8 Integration Decision Making

5.2 Differential Equations

5.2.1 General Solutions

5.2.2 Particular Solutions

5.2.3 Separation of Variables

5.2.4 Modelling with Differential Equations

5.2.5 Solving & Interpreting Differential Equations

6. Parametric Equations

6.1 Parametric Equations

6.1.1 Parametric Equations - Basics

6.1.2 Parametric Equations - Eliminating the Parameter

6.1.3 Parametric Equations - Sketching Graphs

6.2 Further Parametric Equations

6.2.1 Parametric Differentiation

6.2.2 Parametric Integration

6.2.3 Parametric Volumes of Revolution

7. Vectors

7.1 Vectors in 2 Dimensions

7.1.1 Basic Vectors

7.1.2 Magnitude & Direction

7.1.3 Vector Addition

7.1.4 Position Vectors

7.1.5 Problem Solving using Vectors

7.2 Vectors in 3 Dimensions

7.2.1 Vectors in 3 Dimensions

7.2.2 Problem Solving using 3D Vectors

7.3 Further Vectors

7.3.1 Equation of a Line in Vector Form

7.3.2 Parallel, Intersecting & Skew Lines

7.3.3 The Scalar ('Dot') Product

7.3.4 Uses of the Scalar Product

Author: Paul