Calculus AB · Applications of Integration · 14 min read · Updated 2026-05-10
Disc method around the x- or y-axis — AP Calculus AB
AP Calculus AB · Applications of Integration · 14 min read
1. What is the Disc Method?★★☆☆☆⏱ 3 min
The disc method (also spelled disk method, both spellings are accepted on the AP exam) is an integral calculus technique used to find the volume of a solid of revolution: a 3D solid formed when a 2D region bounded by curves is rotated around a horizontal or vertical axis.
The method gets its name from the thin, circular disc-shaped cross-sections we use to approximate the solid’s total volume, analogous to how thin rectangles approximate area under a curve.
2. Disc Method Around the X-Axis★★☆☆☆⏱ 4 min
When rotating a region bounded above by $y = f(x)$, below by the x-axis, between $x=a$ and $x=b$ around the x-axis, we slice perpendicular to the axis of rotation. The x-axis is horizontal, so perpendicular slices are vertical with thickness $dx$. For any $x$ between $a$ and $b$, the radius of the disc at that position is the perpendicular distance from the x-axis to the curve: $r = f(x)$.
V = \pi \int_a^b \left[f(x)\right]^2 dx
This formula comes directly from Riemann sums: as the number of slices approaches infinity, the sum of individual disc volumes becomes the definite integral. The method only works when the region is adjacent to the axis of rotation, with no gap between the region and the axis. A disc is just a special case of a washer with an inner radius of 0.
Exam tip: Always square the entire radius before integrating, not just the variable term. A common mistake is writing $(2\sqrt{x})^2 = 2x$ instead of $4x$, which loses the square of the constant coefficient.
3. Disc Method Around the Y-Axis★★★☆☆⏱ 4 min
When rotating around the vertical y-axis, we slice perpendicular to the axis of rotation, so slices are horizontal with thickness $dy$. This means we must express the radius as a function of $y$, not $x$, and integrate with respect to $y$.
V = \pi \int_c^d \left[g(y)\right]^2 dy
For a region bounded on the right by $x = g(y)$, on the left by the y-axis, between $y = c$ and $y = d$, the radius of each disc is the horizontal perpendicular distance from the y-axis to the curve, so $r = g(y)$. The only key difference from x-axis rotation is rearranging the original function to solve for $x$ in terms of $y$ before setting up the integral.
Exam tip: When rotating around the y-axis, double-check that you are integrating with respect to $y$, not $x$. Even if all your arithmetic is correct, integrating $dx$ for a y-axis rotation will always give the wrong answer.
4. Disc Method Around Shifted Axes★★★☆☆⏱ 3 min
The disc method works for any horizontal or vertical axis of rotation, not just the x-axis ($y=0$) or y-axis ($x=0$). The core principle never changes: the radius $r$ is always the perpendicular distance between the curve and the axis of rotation.
For a horizontal axis of rotation $y = k$, we still integrate with respect to $x$, and the radius is $r = |f(x) - k|$. Squaring eliminates the sign, so $r^2 = (f(x) - k)^2$. For a vertical axis of rotation $x = h$, we integrate with respect to $y$, and $r^2 = (g(y) - h)^2$.
Exam tip: When the axis is shifted below the x-axis or to the right of the y-axis, the distance increases, so don’t forget to add the shift when calculating radius.
5. AP Style Practice★★★★☆⏱ 4 min
Common Pitfalls
Why: Students get used to integrating $dx$ for x-axis rotation and forget to switch variables for y-axis rotation
Why: Students rush squaring and only square the variable term, forgetting to square constant coefficients
Why: Students memorize $r = f(x)$ for x-axis rotation and don’t adapt the formula for shifted axes
Why: Students confuse disc method (for regions adjacent to the axis) with the washer method (for regions with a gap)
Why: Students copy bounds from the problem statement without checking which variable they correspond to