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Volumes of Revolution (CIE A Level Maths: Pure 1)
Revision Note
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 (and other boundary equations) is rotated 360° around the x-axis
- A volume of revolution is the volume of this solid formed
Example of a solid of revolution that is formed by rotating the area bounded by the function , the lines and and the -axis about the -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 , the lines and , and the x-axis is rotated 360° about the x-axis use the formula
- 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)
- The answer may be required in exact form (leave in terms of π)
- Trickier questions may give you the volume and ask for the value of an unknown constant elsewhere in the problem
Worked example
Examiner Tip
- To help remember the formula note that it is only - 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
- 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)
- If rotating a single point around the x-axis a circle of radius would be formed
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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 (and other boundary equations) is rotated 360° around the y-axis
- A volume of revolution is the volume of this solid formed
Example of a solid of revolution that is formed by rotating the area bounded by the function , the lines and and the -axis about the -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 , the lines and , and the y-axis is rotated 360° about the y-axis use the formula
- Note that although the function may be given in the form it will first need rewriting in the form
- 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 into the form (if necessary)
- This is finding the inverse function
- 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)
- The answer may be required in exact form (leave in terms of π)
- Trickier questions may give you the volume and ask for the value of an unknown constant elsewhere in the problem
Worked example
Examiner Tip
- Double check questions to ensure you are clear about which axis the rotation is around
- Separating the rearranging of into 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
- In some cases it can seem as though x has been squared twice
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