Washer method around the x- or y-axis — AP Calculus AB
1. What Is the Washer Method? ★☆☆☆☆ ⏱ 2 min
The washer method is an integration technique used to calculate the volume of a solid of revolution that has a hollow core, formed when rotating a region between two curves around an axis. Each cross-section perpendicular to the axis forms a "washer" (a flat disk with a smaller circular hole cut out of the center), hence the name, and it is an extension of the disk method for solid solids of revolution.
Per the AP Calculus AB Course and Exam Description, this topic is part of Unit 8, which accounts for 10–15% of the total AP exam score. Washer method problems appear on both multiple-choice and free-response sections, and you will often be asked to set up or evaluate integrals for volume on FRQs, making this a high-weight topic.
2. Revolution Around the x-Axis (Horizontal Axes) ★★☆☆☆ ⏱ 4 min
When rotating around any horizontal axis (of the form $y=k$, including the x-axis $y=0$), cross-sections perpendicular to the axis are vertical slices, so we integrate with respect to $x$. The area of one washer is the area of the outer circle minus the area of the inner circle:
A = \pi R^2 - \pi r^2 = \pi\left(R^2 - r^2\right)
where $R(x)$ is the outer radius (distance from the axis to the farther curve from the axis), and $r(x)$ is the inner radius (distance from the axis to the closer curve from the axis). Total volume is the integral of all cross-sectional areas between the bounds of the region:
V = \pi \int_a^b \left(\left[R(x)\right]^2 - \left[r(x)\right]^2\right)dx
For rotation around the x-axis ($y=0$), this simplifies to $R(x) = y_{\text{upper}}(x)$ and $r(x) = y_{\text{lower}}(x)$, since the distance from $y=0$ to a curve is just the $y$-value of the curve.
Exam tip: Always square each radius separately before subtracting. Never subtract first then square, as this will always give the wrong volume.
3. Revolution Around the y-Axis (Vertical Axes) ★★☆☆☆ ⏱ 4 min
When rotating around any vertical axis (of the form $x=k$, including the y-axis $x=0$), cross-sections perpendicular to the axis are horizontal slices, so we integrate with respect to $y$. This means we must first rewrite all bounding curves as functions of $y$, instead of $y$ as a function of $x$. The logic for radii and area is identical to the horizontal case:
V = \pi \int_c^d \left(\left[R(y)\right]^2 - \left[r(y)\right]^2\right)dy
For rotation around the y-axis ($x=0$), $R(y) = x_{\text{right}}(y)$ (farther from the axis) and $r(y) = x_{\text{left}}(y)$ (closer to the axis).
Exam tip: Always rewrite all functions in terms of the integration variable before calculating radii. Mixing $x$ and $y$ in the same integral is an immediate point loss on FRQs.
4. Revolution Around Non-Coordinate Axes ★★★☆☆ ⏱ 4 min
AP Calculus AB exam questions frequently ask for rotation around an axis that is not the x or y axis (e.g. $y=2$, $x=-1$). The core logic stays the same, but you must correctly calculate the radius as the distance between the curve and the shifted axis, not just the value of the curve.
Exam tip: Test a sample point in your interval to confirm $R > r$: if your integrand is negative, you swapped outer and inner radii.
5. AP-Style Concept Check ★★★☆☆ ⏱ 3 min
Common Pitfalls
Why: Students confuse factoring rules with the circle area formula, or rush through setup.
Why: Students forget cross-sections must be perpendicular to the axis of rotation, which dictates the integration variable.
Why: Students get used to rotating around the x-axis ($y=0$) and forget to adjust radii for a shifted axis.
Why: Students mix up which curve is farther from the axis, especially when the axis is above or to the left of the region.
Why: Students remember the core integrand but omit the constant from the circle area formula.
Why: Students are more comfortable with $y$ as a function of $x$ and forget to invert.