Volumes of Revolution About the x-axis
What is a volume of revolution about the x-axis?
- A solid of revolution is formed
- when an area bounded by a function and the lines and
- is rotated radians about 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!
What is the formula for a volume of revolution about the x-axis?
The volume of revolution of a solid rotated radians () about the -axis between and is given by:
-
- This is not given on the exam formula sheet, so you need to remember it
- Note that is the area of the circular cross-section of the solid at any value of
- That might help you remember the form of the volume integral
- is a function of
- i.e.
- and are the equations of the (vertical) lines bounding the area
- ( is the 'left boundary' and is the 'right boundary')
- and may be given in the question
- one boundary may be the -axis ()
- the -axis intercepts of may also be boundaries
- This is not given on the exam formula sheet, so you need to remember it
How do I calculate the volume of revolution about the x-axis?
- STEP 1
Identify the limits and- These may be given in the question
- or be indicated on a graph in the question
- Sketching the graph of can help if the graph is not provided
- These may be given in the question
- STEP 2
Square the function- e.g.
- or
- STEP 3
Evaluate the integral in the volume formula- An answer may be required in exact form
- i.e. as a multiple of
- An answer may be required in exact form
Exam Tip
- Don't panic if involves a square root
- The square root will disappear when you find
- Don't forget to bring back in after working out the integral
- In my experience that is a very common student error
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 about the -axis. Give your answer as an exact value.
Start by finding the values of and for the formula
'Bounded by the coordinate axes' tells us that the -axis () is one boundary
The question tells us that is the other one
So and
If in doubt, drawing a sketch can help
Now square the function
Substitute everything into
Work out the definite integral
Put that value back into the volume formula
Don't forget to include the !
The question asks for an exact value answer, so leave the answer in terms of
Volume