Volumes of Revolution (DP IB Analysis & Approaches (AA): HL): Revision Note

Volumes of revolution around 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 $y=f\left(x\right)$
  (and other boundary equations) is rotated $2\pi$ radians $(360°)$ around the $x$-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

How do I solve problems involving a volume of revolution around x-axis?

• Use the formula

 $V=\pi {\int }_a^b{y}^{2}dx$

• $y$ is a function of $x$

• $x=a$ and $x=b$ are the equations of the (vertical) lines bounding the area

  • If $x=a$ and $x=b$ are not stated in a question, the boundaries could involve the $y$-axis ($x=0$) and/or a root of $y=f\left(x\right)$

  • You can use a GDC to plot the curve, sketch it and highlight the area to help

  • Visualising the solid created is helpful

    • Try sketching some functions and their solids of revolution for practice

• STEP 1
  Identify the limits $a$ and $b$

  • Sketching the graph of $y=f\left(x\right)$ or using a GDC to do so is helpful, especially when $a$ and $b$ are not given directly in the question
     

• STEP 2
  Square the $y$ function
   

• STEP 3
  Use the formula to evaluate the integral and find the volume of revolution

  • An answer may be required in exact form

Volumes of revolution around y-axis

What is a volume of revolution around the y-axis?

• Very similar to above, this is a solid of revolution which is formed when an area bounded by a function $y=f\left(x\right)$ (and other boundary equations) is rotated $2\pi$ radians $(360°)$ around the $y$-axis

• The volume of revolution is the volume of this solid

How do I solve problems involving a volume of revolution around the y-axis?

• Use the formula

$V=\pi {\int }_a^b{x}^{2}dy$ 

• $x$ is a function of $y$

  • the function is usually given in the form $y=f\left(x\right)$

  • this will need rearranging into the form $x=g\left(y\right)$

• $y=a$ and $y=b$ are the equations of the (horizontal) lines bounding the area

  • If $y=a$ and $y=b$ are not stated in the question, the boundaries could involve the $x$-axis ($y=0$) and/or a root of $x=g\left(y\right)$

  • Use a GDC to plot the curve, sketch it and highlight the area for practice

  • Visualising the solid created is helpful

    • Try sketching some functions and their solids of revolution to help

• STEP 1
  Identify the limits $a$ and $b$

  • Sketching the graph of $y=f\left(x\right)$ or using a GDC to do so is helpful, especially if $a$ and $b$ are not given directly in the question

• STEP 2
  Rearrange $y=f\left(x\right)$ into the form $x=g\left(y\right)$

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

• STEP 3
  Square the $x$ function
   

• STEP 4
  Use the formula to evaluate the integral and find the volume of revolution

  • An answer may be required in exact form