# Edexcel A Level Further Maths: Core Pure:复习笔记5.1.1 Volumes of Revolution

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

#### Exam 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)

#### 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  radians around the χ-axis.  Give your answer as an exact multiple of .

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

#### Exam 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

#### 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.

### Volumes of Revolution using Parametric Equations

#### What is parametric volumes of revolution?

• Solids of revolution are formed by rotating functions about the x-axis or the y-axis
• Here though, rather than given y in terms of x, both x and y are given in terms of a parameter, t
• Depending on the nature of the functions f and g it may not be convenient or possible to find y in terms of x

#### How do I find volumes of revolution when x and y are given parametrically?

• The aim is to replace everything in the ‘original’ integral so that it is in terms of t
• For the ‘original’ integral   and parametric equations given in the form  use the following process
• STEP 1: Find dx or dy in terms of t and dt
• STEP 2: If necessary, change the limits from x values or y values to t values using
•
• STEP 3: Square y or x
•
• Do this separately to avoid confusing when putting the integral together
• STEP 4: Set up the integral, so everything is now in terms of t, simplify where possible and evaluate the integral to find the volume of revolution

(if around x-axis) or  (if around y-axis)

#### Exam Tip

• Avoid the temptation to jump straight to STEP 4
• There could be a lot to change and simplify in exam style problems
• Doing each step carefully helps maintain high levels of accuracy

#### Worked Example

The curve C is defined parametrically by  and . C is rotated 360° about the x-axis between the values of  and . Show that the volume of the solid of revolution generated by this rotation is  cubic units where  and  are integers to be found.