Calculus BC · CED Unit 8: Applications of Integration · 14 min read · Updated 2026-05-11
Disc method around other axes — AP Calculus BC
AP Calculus BC · CED Unit 8: Applications of Integration · 14 min read
1. Revolution Around Horizontal Shifted Axes★★☆☆☆⏱ 4 min
A horizontal axis of rotation always has the form $y = k$, where $k$ is a constant. For the disc method, slices are perpendicular to the axis, so we use vertical slices and integrate with respect to $x$. The only adjustment from the basic disc method is calculating the radius correctly: radius is the distance between the function $y = f(x)$ and the axis $y = k$.
V = \pi \int_a^b \left[ f(x) - k \right]^2 dx
Since distance is always positive, and squaring any real number gives a non-negative result, you do not need to keep absolute value after squaring. Shifting the axis only changes the radius, not the integration bounds or variable.
2. Revolution Around Vertical Shifted Axes★★★☆☆⏱ 4 min
A vertical axis of rotation has the form $x = h$, where $h$ is a constant. For the disc method, slices must be perpendicular to the vertical axis, so we use horizontal slices and integrate with respect to $y$. This requires rewriting any function given as $y = f(x)$ into the form $x = g(y)$ first.
V = \pi \int_c^d \left[ g(y) - h \right]^2 dy
The core logic for radius calculation is identical to horizontal shifted axes: radius is just the distance between the function $x = g(y)$ and the axis $x = h$.
3. Radius Adjustment for Regions on the Opposite Side of the Axis★★★☆☆⏱ 3 min
A common AP exam variation has the entire region located below a horizontal axis or to the left of a vertical axis. The core rule for radius does not change: radius is always the positive distance between the region edge and the axis.
For a region entirely below horizontal axis $y=k$: $r = k - f(x)$
For a region entirely above horizontal axis $y=k$: $r = f(x) - k$
For a region entirely left of vertical axis $x=h$: $r = h - g(y)$
For a region entirely right of vertical axis $x=h$: $r = g(y) - h$
Note that $(f(x) - k)^2 = (k - f(x))^2$, so any order of subtraction gives the same volume after squaring, but using a positive radius makes it easier to catch setup errors when checking your work.
4. AP-Style Concept Check★★★★☆⏱ 3 min
Common Pitfalls
Why: Students default to $x$ integration because most functions are written as $y=f(x)$, forgetting disc method requires slices perpendicular to the axis of rotation.
Why: Students memorize the basic formula for the x-axis and forget to shift the radius when the axis moves.
Why: Students associate any solid of revolution around a non-standard axis with the disc method, regardless of whether the solid is hollow.
Why: Students expect the function to be larger than the axis constant, so they get confused when the opposite is true.
Why: Students remember radius is positive, so they keep absolute value out of caution.