Volumes of Revolution (CIE AS Maths: Pure 1)

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Author

Paul

Last updated

23 August 2023

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Volumes of revolution around the x-axis

What is a volume of revolution around the x-axis?

A solid of revolution is formed when an area bounded by a function $y = f (x)$ (and other boundary equations) is rotated 360° around the x-axis
A volume of revolution is the volume of this solid formed

2dk4S6Oy_6-2-4-cie-fig1-vol-of-rev

Example of a solid of revolution that is formed by rotating the area bounded by the function $y = f (x)$ , the lines $x = a$ and $x = b$ and the $x$ -axis $360 °$ about the $x$ -axis

How do I find the volume of revolution around the x-axis?

To find the volume of revolution created when the area bounded by the function $y = f (x)$ , the lines $x = a$ and $x = b$ , and the x-axis is rotated 360° about the x-axis use the formula

$V = π \int_{a}^{b} y^{2} d x$

The formula may look complicated or confusing at first due to the y and dx
- remember that y is a function of x
- once the expression for y is substituted in, everything will be in terms of x
π is a constant so you may see this written either inside or outside the integral
This is not given in the formulae booklet
- The formulae booklet does list the volume formulae for some common 3D solids – it may be possible to use these depending on what information about the solid is available

How do I solve problems involving volumes of revolution around the x-axis?

Visualising the solid created is helpful
- Try sketching some functions and their solids of revolution to help
STEP 1 Square y
- Do this first without worrying about π or the integration and limits
STEP 2 Identify the limits a and b (which could come from a graph)
STEP 3 Use the formula by evaluating the integral and multiplying by π
- The answer may be required in exact form (leave in terms of π)
  - If not, round to three significant figures (unless told otherwise)
Trickier questions may give you the volume and ask for the value of an unknown constant elsewhere in the problem

Worked example

6-2-4-cie-fig2-we-solution

Examiner Tip

To help remember the formula note that it is only $y^{2}$ - volume is 3D so you may have expected a cubic expression
- If rotating a single point around the x-axis a circle of radius would be formed
  - The area of that circle would then be $π y^{2}$
  - Integration then adds up the areas of all circles between a and b creating the third dimension and volume
    (In 2D, integration creates area by adding up lots of 1D lines)

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Volumes of revolution around the y-axis

What is a volume of revolution around the y-axis?

A solid of revolution is formed when an area bounded by a function $y = f (x)$ (and other boundary equations) is rotated 360° around the y-axis
A volume of revolution is the volume of this solid formed

6-2-4-cie-fig3-vol-of-rev-y-axis

Example of a solid of revolution that is formed by rotating the area bounded by the function $y = f (x)$ , the lines $y = c$ and $y = d$ and the $x$ -axis $360 °$ about the $y$ -axis

How do I find the volume of revolution around the y-axis?

To find the volume of revolution created when the area bounded by the function $y = f (x)$ , the lines $y = c$ and $y = d$ , and the y-axis is rotated 360° about the y-axis use the formula

$V = π \int_{c}^{d} x^{2} d y$

Note that although the function may be given in the form $y = f (x)$ it will first need rewriting in the form $x = g (y)$
This is not given in the formulae booklet

How do I solve problems involving volumes of revolution around the y-axis?

Visualising the solid created is helpful
- Try sketching some functions and their solids of revolution to help

STEP 1 Rearrange $y = f (x)$ into the form $x = g (y)$ (if necessary)
- This is finding the inverse function $f^{- 1} (x)$
STEP 2 Square x
- Do this first without worrying about π or the integration and limits
STEP 3 Identify the limits c and d (which could come from a graph)
STEP 4 Use the formula by evaluating the integral and multiplying by π
- The answer may be required in exact form (leave in terms of π)
  - If not, round to three significant figures (unless told otherwise)

Trickier questions may give you the volume and ask for the value of an unknown constant elsewhere in the problem

Worked example

6-2-4-cie-fig4-we-solution-part-1

6-2-4-cie-fig4-we-solution-part-2

Examiner Tip

Double check questions to ensure you are clear about which axis the rotation is around
Separating the rearranging of $y = f (x)$ into $x = g (y)$ and the squaring of x is important for maintaining accuracy
- In some cases it can seem as though x has been squared twice
  - in the worked example above, x has been squared twice
  - but it needed to be – once as part of the rearranging, once as part of the volume formula

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