Modelling with Volumes of Revolution (DP IB Maths: AA HL)

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Last updated

16 July 2024

The volume of the solid of revolution formed by rotating an area through $2 π$ radians around the $x$ -axis is $V = π \int_{a}^{b} y^{2} d x$ , and for the $y$ -axis is $V = π \int_{a}^{b} x^{2} d y$ . These are both given in the formula booklet.

Adding & Subtracting Volumes

When would volumes of revolution need to be added or subtracted?

The ‘curve’ boundary of an area may consist of more than one function of $x$

For example

the ‘curve’ boundary from $x = 0$ to $x = 3$ is $y = f (x)$
the ‘curve’ boundary from $x = 3$ to $x = 6$ is $y = g (x)$

So the total volume would be $V = π \int_{0}^{3} {[f (x)]}^{2} d x + π \int_{3}^{6} {[g (x)]}^{2} d x$

The solid of revolution may have a ‘hole’ in it

e.g. a ‘toilet roll’ shape would be the difference of two cylindrical volumes

How do I know whether to add or subtract volumes of revolution?

When the area to be rotated around the $x$ -axis has more than one function defining its boundary it can be trickier to tell whether to add or subtract volumes of revolution
- It will depend on the nature of the functions and their points of intersection
- With help from a GDC, sketch the graph of the functions and highlight the area required

How do I solve problems involving adding or subtracting volumes of revolution?

Visualising the solid created becomes increasingly useful (but also trickier) for shapes generated by separate volumes of revolution

Continue trying to sketch the functions and their solids of revolution to help

STEP 1

Identify the functions

(y = f (x), y = g (x), . . .)

involved in generating the volume

Determine whether the separate volumes will need to be added or subtracted

Identify the limits for each volume involved

Sketching the graphs of

y = f (x)

and

y = g (x)

, or using a GDC to do so, is helpful, especially when the limits are not given directly in the question

STEP 2

Square

y

for all functions

({[f (x)]}^{2}, {[g (x)]}^{2}, . . .)

This step is not essential if a GDC can be used to calculate integrals and an exact answer is not required.

STEP 3

Use the appropriate volume of revolution formula for each part, evaluate the definite integral and add or subtract as necessary

The answer may be required in exact form

Examiner Tip

A sketch of the graph, limits, etc is always helpful, whether one has been given in the question or not
- Use your GDC where possible

Worked example

Find the volume of revolution of the solid formed by rotating the region enclosed by the positive coordinate axes and the graphs of $y = 2^{x}$ and $y = 4 - 2^{x}$ by $2 π$ radians around the $x$ -axis. Give your answer to three significant figures.

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Modelling with Volumes of Revolution

What is meant by modelling volumes of revolution?

Many everyday objects such as buckets, beakers, vases and lamp shades can be modelled as a solid of revolution
The volume of revolution of the solid can then be calculated
An object that would usually stand upright can be modelled horizontally so its volume of revolution can be found

What modelling assumptions are there with volumes of revolution?

The solids formed are usually the main shape of the body of the object
- For example, the handle on a bucket would not be included
The thickness of the solid is negligible relative to the size of the object
- thickness will depend on the purpose of the object and the material it is made from

How do I solve modelling problems with volumes of revolution?

Visualising and sketching the solid formed can help with starting problems
Familiarity with applying the volume of revolution fomulae
- around the x-axis: $V = \int_{a}^{b} y^{2} d x$
- around the y-axis: $V = \int_{a}^{b} x^{2} d y$
The volume of revolution may involve adding or subtracting partial volumes
Questions may ask related questions in context
- g. A question about a bucket may ask about its capacity
  - this would be measured in litres
  - so a conversion of units may be required
  - (1000 cm³ = 1 litre)

Examiner Tip

Remember to answer questions directly
- In modelling scenarios, interpretation is often needed after finding the 'final answer'
Modelling questions often ask about assumptions, criticisms and/or improvements
Examples
- it is assumed the thickness of the material an object is made from is negligible
- a 'smooth' curve may not be a good model if the item is being made from a rough material
- other things may significantly reduce the volume found and impact conclusions
  - e.g. Stones, plants and decorations placed in an aquarium will reduce the volume of water needed to fill it - and hence the number/size/type of fish it can accommodate may be impacted