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Volumes of Revolution (DP IB Maths: AI HL)
Revision Note
Volumes of Revolution Around 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 radians around the -axis - The volume of revolution is the volume of this solid
- Be careful – the ’front’ and ‘back’ of this solid are flat
- they were created from straight (vertical) lines
- 3D sketches can be misleading
How do I solve problems involving the volume of revolution around the x-axis?
- Use the formula
-
- This is given in the formula booklet
- is a function of
- and are the equations of the (vertical) lines bounding the area
- If and are not stated in a question, the boundaries could involve the -axis () and/or a root of
- Use a GDC to plot the curve, sketch it and highlight the area to help
- Visualising the solid created is helpful
- Try sketching some functions and their solids of revolution to help
If not identifiable from the question, use the graph to find the limits and
Examiner Tip
- Functions involved can be quite complicated so type them into your GDC carefully
- Whether a diagram is given or not, using your GDC to plot the curve, limits, etc (where possible) can help you to visualise and make progress with problems
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 .
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Volumes of Revolution Around y-axis
What is a volume of revolution around the y-axis?
- Very similar to above, this is a solid of revolution which is formed when an area bounded by a function (and other boundary equations) is rotated radians around the -axis
- The volume of revolution is the volume of this solid
How do I solve problems involving the volume of revolution around y-axis?
- Use the formula
-
- This is given in the formula booklet
- is a function of
- the function is usually given in the form
- this will need rearranging into the form
- and are the equations of the (horizontal) lines bounding the area
- If and are not stated in the question, the boundaries could involve the -axis () and/or a root of
- Use a GDC to plot the curve, sketch it and highlight the area to help
- Visualising the solid created is helpful
- Try sketching some functions and their solids of revolution to help
If not identifiable from the question use the graph to find the limits and
Examiner Tip
- Functions involved can be quite complicated so type them into your GDC carefully
- Whether a diagram is given or not, using your GDC to plot the curve, limits, etc (where possible) can help you to visualise and make progress with problems
Worked example
Find the volume of the solid of revolution formed by rotating the region bounded by the graph of and the coordinate axes by radians around the -axis. Give your answer to three significant figures.
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