Modelling with Volumes of Revolution (DP IB Analysis & Approaches (AA)): Revision Note

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Last updated

6 December 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 Tips and Tricks

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.