Disc Method Around the x-Axis (College Board AP® Calculus AB): Revision Note

Volume with disc method revolving 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 $y=f\left(x\right)$
  (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 can I use the disc method to calculate a volume of revolution around the x-axis?

• For a continuous function $f$, if the region bounded by

  • the curve $y=f\left(x\right)$ and the $x$-axis

  • between $x=a$ and $x=b$

• is rotated $2\pi$ radians $(360°)$ around the $x$-axis, then the volume of revolution is

  •  $V={\int }_a^b\pi y^2\mathit{d}x=\pi {\int }_a^b{y}^2\mathit{d}x$

    • Note that $y$ is a function of $x$

• If $x=a$ and $x=b$ are not stated in a question, these boundaries could involve

  • the $y$-axis ($x=0$)

  • and/or an $x$-intercept of $y=f\left(x\right)$

• This method of finding volumes of revolution uses the idea of a definite integral as calculating an accumulation of change

  • It is a special case of 'finding volumes from areas of known cross-sections'

  • $\pi y^2·∆x$ is the volume of a disc with

    • circular cross section of radius $\left|y\right|$

    • and length $∆x$

  • $\pi y^{2}dx$ is the limit of this volume element as $∆x\to 0$

  • The integral ${\int }_a^b\pi y^{2}dx$ sums up all these infinitesimal volume elements between $x=a$ and $x=b$