Calculus AB — volumes of revolution with discs and washers

Revolving a region around a line and finding the volume of the solid it sweeps out is one of the most reliable free-response parts on the AP Calculus AB exam, and a regular multiple-choice item too. In AB the tools are two, the disc method and the washer method (the shell method belongs to other courses), and topics 8.9 to 8.12 of the course framework ask for both around the coordinate axes and around other horizontal and vertical lines. This material trains all four situations.

The twelve quiz questions move from the simplest case to the ones that cost points. You revolve the region under the square-root curve about the x-axis, and a region bounded by x = y² about the y-axis, where the slices are horizontal and everything has to be written in y. You then meet washers: the region between y = x and y = x² about the x-axis, about the y-axis with the curves rewritten as x = y and x = √y, and about the line y = −1 below the region, where one unit has to be added to both radii. Three questions ask only for the correct set-up, among them one about the line y = 2 above a region and one about the vertical line x = 1, because choosing the outer and inner radius is where most mistakes happen. A further question names the classic error, squaring the difference of the radii instead of subtracting the squares. The last items revolve a region about the line x = 4 at its own edge, which gives discs of radius 4 − y², and fill a bowl, measured in centimeters, whose inner surface is a parabola turned about its axis.

The wrong options are the values those mistakes really produce: the volume about the wrong axis, a radius measured from the y-axis instead of from the line of revolution, a forgotten hole, a radius left unsquared, a shift added to only one radius. Each explanation writes out the radii, the limits and the integral, and names the error behind the most tempting wrong value.

The flashcards hold the disc and washer formulas, the definitions of outer and inner radius, the rule for choosing dx or dy from the direction of the axis, and the radii about a line y = k above or below the region.

This is independent practice written by Zestly, based on the published AP Calculus AB course framework; it is not produced or endorsed by the College Board.

  • Set up and evaluate disc volumes about the x-axis and the y-axis
  • Set up and evaluate washer volumes with outer and inner radii
  • Measure radii from a horizontal or vertical line other than an axis
  • Choose integration in x or in y from the direction of the axis of revolution
  • Recognize the error of squaring the difference of the radii
  • Apply a volume of revolution to a container described in context

Practice material written by Zestly, based on the College Board AP Calculus AB course framework (Unit 8: Applications of Integration, topics 8.9–8.12: volume with the disc and washer methods revolving around the x- or y-axis and around other axes), 2026–27 course and exam description.

Sample question

A region bounded by $y = x^2$ and $y = 0$ for $0 \le x \le 1$ is revolved about $y = 2$. What is the correct integral setup?

See the answer

$\pi \int_{0}^{1} (2^2 - (2-x^2)^2) dx$

The outer radius is the distance from $y=2$ to $y=0$, which is $2-0=2$. The inner radius is the distance from $y=2$ to $y=x^2$, which is $2-x^2$. $V = \pi \int (R^2 - r^2) dx$.

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