Volumes of Revolution (Cambridge International AS Mathematics)

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

$V = \pi \int_{a}^{b} y^{2} d x$

• 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

Worked Example

Exam Hint

• To help remember the formula note that it is only $y^{2}$ - 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 $πy^{2}$
    • 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)

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

$V = \pi \int_{c}^{d} x^{2} d y$

• Note that although the function may be given in the form $y=f(x)$ it will first need rewriting in the form $x=g(y)$
• 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 $y=f(x)$ into the form $x=g(y)$ (if necessary)

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

• 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

Worked Example

Exam Hint

• Double check questions to ensure you are clear about which axis the rotation is around

• Separating the rearranging of $y=f(x)$into $x=g(y)$ and the squaring of x is important for maintaining accuracy

  • In some cases it can seem as though *x *has been squared twice

    • in the worked example above, *x *has been squared twice
    • but it needed to be – once as part of the rearranging, once as part of the volume formula