Key takeaways
- Shell method: V = 2π ∫ₐᵇ x · f(x) dx for spinning around the y-axis.
- Each shell unrolls into a thin slab: 2πx long, f(x) tall, dx thick.
- Don't forget the x inside the integral. It is the shell's radius.
- Shells avoid solving y = f(x) for x when you spin around the y-axis.
In this guide
What is the shell method?
Take the region under y = f(x) and spin it around the y-axis. Instead of slicing the solid into coins, think of it as many thin, hollow tubes nested inside each other, like cups stacked one inside another. Each tube is a cylindrical shell.
Each shell comes from a thin vertical strip at distance x from the axis. Cut one shell open and unroll it: it becomes an almost flat slab. Its length is the distance around, 2πx. Its height is f(x). Its thickness is dx. So its volume is 2πx × f(x) × dx. Add up all the shells and you get the volume.
Why does the shell method matter?
The shell method solves a real headache. To spin y = x² around the y-axis with disks, you'd first need to rewrite it as x = √y and change the limits. Shells let you keep the curve exactly as it is and integrate in x.
It also gives a second way to check any volume. Disks and shells set up correctly always agree, so if you get two different answers, you know one setup is wrong.
Unroll one thin shell and it's just a flat slab: 2πx long, f(x) high and dx thick. That single picture is the whole formula.
The formula
| Symbol | What it means | Example |
|---|---|---|
| V | The volume, in cubic units | 25.13 m³ |
| x | Radius of each shell (its distance from the y-axis) | 1.5 m |
| f(x) | Height of each shell. Here f(x) = xⁿ | x² |
| a, b | Inner and outer edges of the region (need 0 ≤ a < b) | 0 and 2 |
| 2π | Turns a radius into a circumference | 6.2832 |
This calculator uses f(x) = xⁿ. Then x · f(x) = xⁿ⁺¹, and it integrates exactly: V = 2π × [xⁿ⁺² ÷ (n + 2)] from a to b.
How to use the shell method step by step
- 1Find the radius of a shell. It is x, the distance from the y-axis.
- 2Find the height of a shell. It is f(x).
- 3Multiply them. Write
x · f(x). - 4Integrate from a to b. Use the power rule.
- 5Multiply by 2π. Give the answer in cubic units.
Now try your own curve. Enter a power and the interval below and the calculator shows each step.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Using π instead of 2π. The 2π comes from the shell's circumference, 2πr.
- Squaring f(x). That belongs to the disk method, not shells.
- Spinning around the wrong axis. This version of the formula is for rotation around the y-axis.
- Using a negative inner edge. The radius x is a distance from the axis, so keep 0 ≤ a < b.
Tips and tricks
- Quick check: spinning y = x from 0 to 1 around the y-axis gives
2π/3 ≈ 2.0944. - Pick the easier method: slices parallel to the axis → shells; slices across the axis → disks.
- Need the flat area first? Use the area between curves calculator.
Where you'll use it in real life
- Tanks and containers: upright tanks and funnels whose side is given as y = f(x).
- Manufacturing: parts made on a lathe that spin around a vertical axis.
- Design: vases and bowls modelled in 3D software by spinning a profile curve.
- Physics: mass of round objects whose density changes with distance from the centre.
Quick summary
The shell method builds a solid from thin tubes, each with volume 2πx · f(x) · dx. Multiply radius by height, integrate from a to b and multiply by 2π: V = 2π∫x·f(x) dx. Use the calculator above for any curve y = xⁿ spun around the y-axis.
Worked example
An engineer designs a water tank by spinning the region under y = x² (from x = 0 to x = 2, in metres) around the y-axis. What is the volume of that solid? Use n = 2, a = 0 and b = 2.
- 01Revolve f(x) = x^2 around y-axis on [0, 2] using shell method
- 02V = 2π ∫[0,2] x · x^2 dx = 2π ∫[0,2] x^3 dx
- 03= 2π [x^4 / 4]|[0,2]
- 04= 2π (4 − 0)
- 05= 2π × 4
- 06= 25.1327
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the difference between the shell method and the disk method?
Disks are slices cut across the axis of rotation, with volume πr² × thickness. Shells are thin tubes parallel to the axis, with volume 2π × radius × height × thickness. Both give the same answer when set up correctly.
When is the shell method easier?
When you spin around the y-axis but your curve is written as y = f(x). Shells let you integrate in x directly, without solving for x in terms of y.
Why is there 2π in the shell formula but only π in the disk formula?
The shell's volume uses its circumference, 2πr. The disk's volume uses a circle's area, πr².
What is the volume when y = x² from 0 to 2 spins around the y-axis?
V = 2π∫₀² x³ dx = 2π × 4 = 8π ≈ 25.13 cubic units.
