Calculus I › Integrals › full formula sheet
The shell method
Spin vertical slices around a vertical axis — they unroll into cylindrical shells. Radius times height, times 2π.
Notation on this page: radius = distance from the slice to the axis; height = top(x) − bottom(x). Slices are parallel to the axis.
Before this lesson: Disk method · Area between curves
Where it comes from
The problem: revolve the region under y = x² on [0, 2] around the y-axis. Disks would need horizontal slices — and the radius as a function of y, x = √y, with the region’s left edge at x = 0… doable, but awkward. Instead, keep the natural vertical slices: each one, revolved, sweeps out a hollow cylinder — a shell, like a paper-towel tube.
Unroll one shell mentally: cut it vertically and flatten it. You get a slab whose length is the circumference 2π·radius, whose height is the slice’s height, whose thickness is dx:
Before reading on: unroll a thin cylindrical shell of radius r, height h, thickness dx into a flat slab. Before reading: what are the slab’s three dimensions — and its volume?
The judgment call — shells vs disks — is about slice direction vs axis direction:
Practical rule: pick the method that keeps the integrand in the original variable. If revolving around the y-axis would force you to invert y = x² into x = √y for disks, shells let you stay in x.
Derivation
Partition [a, b] into vertical strips; revolve each strip; unroll each tube into a slab; sum and limit.
Radius and height, precisely: radius = distance from the slice (at position x) to the axis — for the y-axis, radius = x; for x = −1, radius = x+1; for x = 4 (region left of it), radius = 4−x. Height = top(x) − bottom(x), exactly as in area-between-curves.
How to use it
Before reading on: vertical slices revolved about the y-axis make shells; about the x-axis they make washers. Before reading: for the region between y = x and y = x2 on [0,1] revolved about the y-axis, which does a vertical slice give — and why?
The procedure, every time:
- Check the geometry: slices parallel to the axis → shells. (If perpendicular is easier, use disks/washers instead — both give the same volume.)
- Radius = distance from slice to axis. y-axis → x. Line x = −1 → x+1. Line x = 4 (region to its left) → 4−x. Always a distance: non-negative.
- Height = top(x) − bottom(x). For a region under one curve above the axis, height = f(x).
- Compute 2π∫(radius)(height)dx. Keep the 2π outside.
- Cross-check with washers when feasible — two methods, one volume. (Example 1 below does exactly this.)
Worked examples
Four problems, easiest first. In each one, read every step — the why of each move is the lesson.
Example 1 — the basic move: y = x² on [0, 2], about the y-axis
- Radius: distance from slice x to the y-axis: radius = x. Height: x² − 0 = x².
- Set up: V = 2π∫02 x·x² dx = 2π∫02 x³ dx.
- Integrate: 2π[x⁴/4]02 = 2π·4 = 8π.
- Cross-check with washers (dy): at height y, x runs √y→2: V = π∫04 (4−y) dy = π[4y − y²/2]04 = π(16−8) = 8π. Matches ✓ — two methods, one volume.
Your turn: y = x2 on [0, 1], about the y-axis. Find V.
Answer: π/2
Radius x, height x2: V = 2π∫01 x·x2 dx = 2π∫01 x3 dx = 2π/4 = π/2. Cross-check with washers (dy): at height y, x runs √y→1: V = π∫01 (1−y) dy = π/2. ✓
Example 2 — cylinder minus cone: y = x on [0, 3], about the y-axis
- Radius = x, height = x. V = 2π∫03 x·x dx = 2π∫03 x² dx.
- Integrate: 2π[x³/3]03 = 2π·9 = 18π.
- Geometry check: at height y, x runs y→3 — the solid is a cylinder (r = 3, h = 3, volume 27π) minus a cone (r = 3, h = 3, volume 9π): 27π − 9π = 18π. Matches ✓
Your turn: y = x on [0, 2], about the y-axis. Find V.
Answer: 16π/3
Radius x, height x: V = 2π∫02 x2 dx = 2π[x3/3]02 = 16π/3. Geometry: cylinder (r = 2, h = 2, volume 8π) minus cone (r = 2, h = 2, volume 8π/3): 8π − 8π/3 = 16π/3. ✓
Example 3 — shifted axis: y = x² on [0, 1], about the line x = −1
- Radius: distance from slice x to x = −1: radius = x − (−1) = x + 1. Height = x².
- Set up: V = 2π∫01 (x+1)·x² dx = 2π∫01 (x³+x²) dx.
- Integrate: 2π[x⁴/4 + x³/3]01 = 2π(1/4 + 1/3) = 2π(7/12) = 7π/6.
- Sanity check: shifting the axis away from the region fattens every shell (radii 1→2 instead of 0→1) — 7π/6 ≈ 3.67 vs the about-y-axis version π/2 ≈ 1.57. Bigger, as expected. ✓
Your turn: y = x on [0, 1], about the line x = −1. Find V.
Answer: 5π/3
Radius: distance from slice x to x = −1: x + 1. Height x. Set up: V = 2π∫01 (x+1)x dx = 2π∫01 (x2+x) dx = 2π[x3/3 + x2/2]01 = 2π(5/6) = 5π/3. Sanity: radii 1→2 (fatter than about the y-axis), so 5π/3 ≈ 5.24 > 2π/3 ≈ 2.09. ✓
Example 4 — height is top−bottom: region between y = x and y = x² on [0, 1], about the y-axis
- Radius = x. Height = top − bottom = x − x² (the area-between-curves gap).
- Set up: V = 2π∫01 x(x−x²) dx = 2π∫01 (x²−x³) dx.
- Integrate: 2π[x³/3 − x⁴/4]01 = 2π(1/3 − 1/4) = 2π(1/12) = π/6.
- Cross-check with washers (dy): at height y, x runs y→√y: V = π∫01 ((√y)² − y²) dy = π∫01 (y−y²) dy = π(1/2 − 1/3) = π/6. Matches ✓
Your turn: Region between y = x2 and y = x3 on [0, 1], about the y-axis. Find V.
Answer: π/10
Radius x; height = top − bottom = x2 − x3. Set up: V = 2π∫01 x(x2−x3) dx = 2π∫01 (x3−x4) dx = 2π[x4/4 − x5/5]01 = 2π(1/20) = π/10. Sanity: a thinner region than the x-vs-x2 one, so π/10 < π/6. ✓
Memorization tips
- Say it aloud: “two-pi radius height dee-x.” All four factors, in order — forget one and the units break.
- Parallel → shells, perpendicular → disks: hold your hand parallel to the axis (tube) vs across it (disk). The gesture is the judgment call.
- Radius is a distance: always measured to the actual axis. Axis x = a → |x−a| with the correct sign for your region.
- Height is top−bottom: the area-between-curves gap rides again. One curve’s height is not the region’s height.
- The unroll picture: circumference × height × thickness. If the 2π ever feels mysterious, unroll the tube.
- Cross-check with washers: when both methods are feasible, they must agree. Two roads, one volume — use the second as a check on the first.
Final challenge
Five mixed questions — shifted axes, the height trap, and shells-vs-disks judgment. Score 5/5 and shells are yours.
← Back to the Calculus I formula sheet
How to learn a formula here
- Read each section in order — every section ends with a short quiz. Take it before moving on; the questions test exactly what you just read.
- Work the examples with the answers covered, then uncover one step at a time and compare.
- Finish with the final challenge — five mixed questions including the classic traps.
- Retake what you miss — every quiz reshuffles each attempt, and every answer explains itself.
Frequently asked questions
What is the shell method?
Revolve vertical slices around a vertical axis: each slice becomes a cylindrical shell of radius = distance to the axis, height = top − bottom, thickness dx, volume 2π·radius·height·dx. Integrate: V = 2π∫ab(radius)(height) dx.
When do I use shells instead of disks?
Shells when the slices are parallel to the axis of rotation (vertical slices + vertical axis); disks/washers when slices are perpendicular. Pick whichever keeps the integrand in the original variable.
Where does the 2π come from?
Unroll the shell: it’s a slab with length = circumference 2π·radius, height = height, thickness = dx. Volume = 2π·radius·height·dx.
What is the radius when the axis is x = −1?
radius = x − (−1) = x + 1: the distance from the slice at position x to the axis. For axis x = a, radius = x − a (region right of axis) or a − x (region left of axis).
What is the height of the shell?
top(x) − bottom(x), exactly as in area-between-curves. For a region under y = f(x) above the x-axis, height = f(x).
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