Disc Method Around Other Axes (College Board AP® Calculus BC): Study Guide

Volume with disc method revolving around other axes

How can I use the disc method to calculate volumes of revolution around axes parallel to the x-axis?

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

  • the curve $y=f\left(x\right)$ and the line $y=k$

  • between $x=a$ and $x=b$

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

  •  $V=\pi {\int }_a^b{\left(y-k\right)}^2\mathit{d}x$

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

• Thinking of this as an accumulation of change

  • $\pi {\left(y-k\right)}^2·∆x$ is the volume of a disc with

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

    • and length $∆x$

  • $\pi {\left(y-k\right)}^{2}dx$ is the limit of this volume element as $∆x\to 0$

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

How can I use the disc method to calculate volumes of revolution around axes parallel to the y-axis?

• This is similar to finding volumes of revolution around axes parallel to the $x$-axis

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

  • the curve $y=f\left(x\right)$ and the line $x=k$

  • between $y=a$ and $y=b$

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

  •  $V=\pi {\int }_a^b{\left(x-k\right)}^2\mathit{d}y$

    • 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 $y$