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

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

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)