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

Volume with disc method revolving around the y-axis

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

• This is very similar to 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 $y$-axis

• The volume of revolution is the volume of this solid

How can I use the disc method to calculate a volume of revolution around the y-axis?

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

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

  • between $y=a$ and $y=b$

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

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

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

      • This will mean rewriting $y=f\left(x\right)$ in the form $x=g\left(y\right)$

    • Also note that the integration is done with respect to $y$

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

  • the $x$-axis ($y=0$)

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