AP Calculus AB and BC

Disk Method vs Shell Method

Slice perpendicular to the axis of rotation and you get disks; slice parallel to it and you get shells. Both return the same volume, so the choice is practical: take the direction whose slices need no rewriting of the curves, because the thickness variable is the one your functions are already solved for.

Disk method

Use when: Slices perpendicular to the axis are already expressed in the variable you would integrate in, and the region meets the axis with no gap.

Shell method

Use when: A perpendicular slice would force you to solve the curves for the other variable, or the boundary of the region changes formula partway along the axis.

Side by side

	Disk method	Shell method
Slice direction	Perpendicular to the axis	Parallel to the axis
Slice shape	A flat circle of radius $r$	A hollow cylinder of radius $r$ and height $h$
Integrand	$\pi r^2$	$2\pi r h$
Rotating about the $x$ axis	Integrate in $x$	Integrate in $y$
When to reach for it	The curves are already solved for the integrating variable	Inverting the curves would be the only alternative
Common trap	Squaring only part of the radius, or losing the $\pi$	Dropping the height $h$ , or measuring $r$ from the wrong line

Fix the slice and the integral follows. A slice perpendicular to the axis of rotation sweeps out a flat circle, so its volume is $\pi r^2$ times the thickness. A slice parallel to the axis sweeps out a cylindrical shell, and unrolling that shell gives a thin sheet of area $2\pi r h$ , again times the thickness. Written out for rotation about a horizontal axis, the disk integral runs in $x$ and the shell integral runs in $y$ .

V = \int_a^b \pi\left[R(x)\right]^2\,dx \qquad \text{versus} \qquad V = \int_c^d 2\pi\,r(y)\,h(y)\,dy

Rotating about a vertical axis swaps those variables. Take the region under $y = x^2$ from $x = 0$ to $x = 2$ , rotated about the $y$ axis. Disks would need slices perpendicular to that axis, so you would rewrite the curve as $x = \sqrt{y}$ and subtract it from an outer radius of $2$ , turning every disk into a washer. Shells keep the original form: $V = \int_0^2 2\pi x \cdot x^2\,dx = 8\pi$ .

The radius is a distance, not a coordinate

Move the axis and only the radius changes. Rotating that same region about the line $x = 3$ makes the shell radius $3 - x$ rather than $x$ , because the radius always measures from the slice to the axis of rotation.

Frequently asked questions

Is the shell method required on the AP exam?

No. The CED lists disks and washers, so every released volume question can be done with them. Shells remain fully valid on free response and are frequently the shorter path.

When does a disk become a washer?

The moment the region stops touching the axis of rotation. A gap leaves a hole in the slice, so you subtract the inner radius and integrate $\pi\left(R^2 - r^2\right)$ .

In the CED: Unit 8: Applications of Integration