Washer method around the x- or y-axis — AP Calculus BC
1. Core Concept of the Washer Method ★★☆☆☆ ⏱ 2 min
The washer method is an integration technique to find the volume of a solid of revolution formed when a region between two curves is rotated around a horizontal or vertical axis. When the rotated region does not touch the axis of rotation, a cross-section cut perpendicular to the axis is a washer: a circular disk with a smaller concentric hole, giving the method its name.
It differs from the simpler disk method, which applies when the region touches the axis (leaving no hole), by accounting for the empty volume between the region's inner boundary and the rotation axis. This topic falls within Unit 8, which counts for 10-15% of your total AP Calculus BC exam score.
2. Washer Method for Rotation Around the x-Axis ★★☆☆☆ ⏱ 4 min
When rotating around the horizontal x-axis, we cut cross-sections perpendicular to the x-axis, so we integrate with respect to $x$. For a region bounded above by $y = R(x)$ (farther from the x-axis) and below by $y = r(x)$ (closer to the x-axis) between $x=a$ and $x=b$, the area of a single washer at position $x$ is the area of the outer circle minus the area of the inner circle:
A(x) = \pi [R(x)]^2 - \pi [r(x)]^2 = \pi\left([R(x)]^2 - [r(x)]^2\right)
Integrating the area across the full interval of $x$ gives the total volume formula:
V = \pi \int_a^b \left( [R(x)]^2 - [r(x)]^2 \right) dx
Exam tip: Always confirm which function is farther from the axis of rotation by testing a value in the interval between intersections; do not assume higher-degree functions are always inner or outer.
3. Washer Method for Rotation Around the y-Axis ★★★☆☆ ⏱ 4 min
When rotating around the vertical y-axis, we cut cross-sections perpendicular to the y-axis, so we must express all boundaries as $x$ in terms of $y$, and integrate over the interval of $y$ values. For a region bounded right by $x=R(y)$ (farther from the y-axis) and left by $x=r(y)$ (closer to the y-axis) between $y=c$ and $y=d$, the volume formula is:
V = \pi \int_c^d \left( [R(y)]^2 - [r(y)]^2 \right) dy
The only change from rotation around the x-axis is that we integrate with respect to $y$. If the AP exam explicitly asks for the washer method, you must use this approach even if the shells method is computationally easier.
Exam tip: If your original functions are given as $y = f(x)$, do not forget to solve explicitly for $x$ in terms of $y$ before setting up the integral; integrating with respect to $x$ for the washer method around a vertical axis is always incorrect.
4. Washer Method for Shifted Axes of Rotation ★★★★☆ ⏱ 5 min
The washer method generalizes to any horizontal or vertical axis of rotation, not just the x-axis ($y=0$) or y-axis ($x=0$). The core rule for radii stays the same: the radius of any boundary curve is the absolute distance between the curve and the axis of rotation, regardless of where the axis is located.
For a horizontal axis at $y=k$, we still integrate with respect to $x$, with radii $R(x) = |\text{outer boundary} - k|$ and $r(x) = |\text{inner boundary} - k|$. For a vertical axis at $x=h$, we integrate with respect to $y$, with radii $R(y) = |\text{outer boundary} - h|$ and $r(y) = |\text{inner boundary} - h|$. The volume formula remains the same after substituting these radii.
Exam tip: Shifted axes are one of the most commonly tested variations on AP FRQ. Always calculate the distance from the axis to every boundary before identifying inner and outer radii.
Common Pitfalls
Why: You confuse the washer method with the shell method, which integrates with respect to $x$ for vertical rotation axes
Why: You incorrectly factor the square when subtracting the areas of the two circles
Why: You assume the inner radius is still measured from the origin, not the new axis of rotation
Why: You do not check if the region boundary changes across the integration interval
Why: You confuse the washer and disk methods; there is no hole so you do not need to subtract an inner area