Volumes of Revolution (Cambridge (CIE) A Level Maths: Pure 1): Revision Note

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 $y=\mathrm{f}\left(x\right)$ (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 $y=f\left(x\right)$ , the lines $x=a$  and  $x=b$ and the $x$-axis $360°$ about the $x$-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 $y=\mathrm{f}\left(x\right)$, the lines $x=a$ and $x=b$, and the x-axis is rotated 360° about the x-axis use the formula

 $V=\pi {\int }_a^b{y}^{2}dx$

• 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

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

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 $y=\mathrm{f}\left(x\right)$ (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 $y=f\left(x\right)$ , the lines $y=c$ and  $y=d$and the $x$-axis $360°$ about the $y$-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 $y=\mathrm{f}\left(x\right)$, the lines $y=c$ and $y=d$, and the y-axis is rotated 360° about the y-axis use the formula

 $V=\pi {\int }_c^d{x}^{2}dy$

• Note that although the function may be given in the form $y=\mathrm{f}\left(x\right)$ it will first need rewriting in the form $x=\mathrm{g}\left(y\right)$ 

• 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 $y=\mathrm{f}\left(x\right)$ into the form $x=\mathrm{g}\left(y\right)$ (if necessary)

  • This is finding the inverse function ${\mathrm{f}}^{-1}\left(x\right)$

• 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