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

Volume with washer method revolving around the x-axis

When should I use the washer method for a volume of revolution around the x-axis?

• The washer method should be used when there is a gap between a region being rotated and the line it is being rotated around

• For example, in the following diagram consider rotating the shaded area between the two curves around the $x$-axis

  • There would be a gap between the $x$-axis and the curve $y_1=f\left(x\right)$

• At a point $x$ between $a$ and $b$, the cross section of the solid of revolution would look like this:

• I.e., it would have the shape of a washer, with

  • $\mathrm{Area}=\pi \left({\left(g\left(x\right)\right)}^2-{\left(f\left(x\right)\right)}^2\right)$

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

• Let $f$ and $g$ be continuous functions such that $\left|f\left(x\right)\right|<\left|g\left(x\right)\right|$ on the interval $\left[a,b\right]$

  • I.e. $f\left(x\right)$ is closer to the $x$-axis than $g\left(x\right)$ is on that interval

• Consider the region bounded by

  • the curves $y_1=f\left(x\right)$ and $y_2=g\left(x\right)$

  • between $x=a$ and $x=b$

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

  •  $V={\int }_a^b\pi \left({y_2}^2-{y_1}^2\right)\mathit{d}x=\pi {\int }_a^b\left({y_2}^2-{y_1}^2\right)\mathit{d}x$

    • Note that $y_1$ and $y_2$ are both functions of $x$

• Make sure that $y_2$ is the curve further away from the $x$-axis

  • and $y_1$ is the curve closer to the $x$-axis

    • If the curves 'swap places' over the interval, then split the calculation into separate integrals

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

  • the $y$-axis ($x=0$)

  • and/or point(s) of intersection of the two curves

• 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'

  • $\pi \left({y_2}^2-{y_1}^2\right)·∆x$ is the volume of a washer with

    • inner radius $\left|y_1\right|$

    • outer radius $\left|y_2\right|$

    • and length $∆x$

  • $\pi \left({y_2}^2-{y_1}^2\right)dx$ is the limit of this volume element as $∆x\to 0$

  • The integral ${\int }_a^b\pi \left({y_2}^2-{y_1}^2\right)dx$ sums up all these infinitesimal volume elements between $x=a$ and $x=b$