Key takeaways
- Disk method: V = π ∫ₐᵇ [f(x)]² dx for spinning around the x-axis.
- Each slice is a thin coin with volume π × radius² × thickness.
- Square f(x) first, then integrate. Never square the finished integral.
- Shapes with a hole need washers: π∫(R² − r²) dx.
In this guide
What is the disk method?
Take the region under a curve y = f(x) and spin it around the x-axis. It sweeps out a solid, like a bowl, a vase or a cone lying on its side. That shape is called a solid of revolution.
Now slice the solid like a loaf of bread, with cuts straight across the x-axis. Each slice is a thin round disk, like a coin. Its radius is the height of the curve, f(x), and its thickness is a tiny dx. A coin's volume is π × radius² × thickness, just like the cylinder volume formula. Add up all the coins and you get the volume.
Why does the disk method matter?
Many everyday objects are round: glasses, bottles, bowls, lamp bases and machine parts made on a lathe. Their volume decides how much they hold, how much material they need and how heavy they are. The disk method gives that volume from just the outline of the shape.
It also shows why round shapes grow so fast. The radius is squared, so a slice twice as wide holds four times as much.
Spin the line y = x from 0 to 1 and you get a cone with radius 1 and height 1. The disk method gives π/3 ≈ 1.0472, exactly the cone formula πr²h ÷ 3.
The formula
| Symbol | What it means | Example |
|---|---|---|
| V | The volume, in cubic units | 25.13 m³ |
| f(x) | Radius of each disk (height of the curve). Here f(x) = xⁿ | √x = x^0.5 |
| a, b | Where the solid starts and ends on the x-axis (need 0 ≤ a < b) | 0 and 4 |
| π | Comes from the area of a circle, πr² | 3.14159 |
This calculator uses f(x) = xⁿ, so [f(x)]² = x²ⁿ, and it integrates this exactly: V = π × [x²ⁿ⁺¹ ÷ (2n + 1)] from a to b.
How to use the disk method step by step
- 1Find the radius. It is the distance from the axis to the curve, f(x).
- 2Square it. Write
[f(x)]². - 3Integrate from a to b. Use the power rule: xᵏ becomes
xᵏ⁺¹ ÷ (k + 1). - 4Multiply by π. Give the answer in cubic units.
Now try your own shape. Enter a power and the start and end points below to see the volume step by step.
Try it yourself
Pre-filled with the example — change any value.
Common mistakes to avoid
- Squaring after integrating.
π(∫f dx)²is wrong. Square first, then integrate. - Using disks for a hollow solid. If there is a hole in the middle, use washers:
π∫(R² − r²) dx. - Using the wrong axis. This formula is for spinning around the x-axis. For the y-axis, the shell method is often easier.
- Forgetting cubic units. Volume is in m³ or cm³, not m².
Tips and tricks
- Convert to litres: 1 m³ = 1,000 litres, and 1,000 cm³ = 1 litre.
- Estimate: the volume must be less than the cylinder with the largest radius,
π × (max radius)² × length. - Check a cone: compare with the cone volume calculator using πr²h ÷ 3.
Where you'll use it in real life
- Kitchen and home: how much a curved bowl, glass or vase holds.
- Manufacturing: the volume, and so the weight, of parts turned on a lathe.
- Aerospace: the volume of a rocket nose cone or fuel tank.
- Construction: the concrete needed for a curved planter or column.
Quick summary
The disk method slices a solid of revolution into thin coins with volume π × radius² × thickness. Square the curve, integrate from a to b and multiply by π: V = π∫[f(x)]² dx. Use the calculator above for any curve y = xⁿ spun around the x-axis.
Worked example
A large concrete planter is shaped by spinning y = √x around the x-axis from x = 0 to x = 4 (in metres). What is the volume of the solid shape? Since √x = x^0.5, use n = 0.5, a = 0 and b = 4.
- 01Revolve f(x) = x^0.5 around x-axis on [0, 4]
- 02V = π ∫[0,4] [x^0.5]² dx = π ∫[0,4] x^1 dx
- 03= π [x^2 / 2]|[0,4]
- 04= π (8 − 0)
- 05= π × 8
- 06= 25.1327
These numbers are pre-filled in the calculator above.
Frequently asked questions
What is the difference between the disk and washer methods?
The disk method is for solid shapes with no hole: V = π∫R² dx. The washer method is for shapes with a hole in the middle: V = π∫(R² − r²) dx, where r is the inner radius.
When should I use the disk method instead of the shell method?
Use disks when your slices are perpendicular to the axis of rotation and the radius is easy to write as a function of the same variable. Shells are often easier when spinning around the y-axis.
Why is there a π in the formula?
Each slice is a circle, and the area of a circle is πr². The π comes from that circle area.
What is the volume when y = √x spins from 0 to 4?
The radius squared is x, so V = π∫₀⁴ x dx = π × 8 ≈ 25.13 cubic units.
