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
Where does the formula for the volume of revolution come from?
- When you integrate to find the area under a curve you can see the formula by splitting the area into rectangles with small widths
- The same method works for volumes
- Split the volume into cylinders with small widths
- The radius will be the y value
- The width will be a small interval along the x-axis δx
- The volume can be approximated by the sum of the volumes of these cylinders
- The limit as δx goes to zero can be found by integration - just like with areas
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
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
Worked example
Find the volume of the solid of revolution formed by rotating the region bounded by the graph of , the coordinate axes and the line by radians around the -axis. Give your answer as an exact multiple of .