When a region under a curve is rotated through 360° about the x-axis, the volume of the solid formed is V = π∫ y² dx between the limits. Square y first, then integrate. For rotation about the y-axis, use V = π∫ x² dy with y-limits. For a region between two curves, subtract the inner solid from the outer one.
Volume of revolution formulae
- About the x-axis: V = π∫ y² dx from a to b
- About the y-axis: V = π∫ x² dy from c to d
- Between two curves (about the x-axis): V = π∫ (y outer² − y inner²) dx
Square first, then integrate. Do not forget the π.
Where does V = π∫ y² dx come from?
Imagine slicing the solid into thin discs perpendicular to the x-axis. Each disc has radius y and thickness δx, so its volume is about πy²δx. Adding them up and letting δx shrink gives V = π∫ y² dx. The method is: square y, set up the integral with π outside, integrate, then evaluate between the limits.
Squaring the curve carefully
Squaring often simplifies a root, because (√f(x))² = f(x). But it expands a sum: (x² + 1)² = x⁴ + 2x² + 1, not x⁴ + 1. Expand fully before integrating. If y has two terms, square the whole of y; never square each term separately.
Rotation about the y-axis
For rotation about the y-axis, the discs are horizontal and their radius is x. Rearrange the equation to give x² in terms of y, use V = π∫ x² dy, and take the limits as y-values.
Trig curves
Squaring a trig function gives sin²x or cos²x, which must be rewritten with a double angle identity before integrating: sin²x = (1 − cos 2x)/2 and cos²x = (1 + cos 2x)/2. For y = tan x, use tan²x = sec²x − 1. Volumes involving trig curves often come out as multiples of π².
Regions between two curves
When the region between two curves is rotated, the solid has a hole. Find the volume swept by the outer curve and subtract the volume swept by the inner one: V = π∫ (y outer² − y inner²) dx. Square each curve separately; (y outer − y inner)² is wrong.
Worked examples
Example 1. The region under y = x from x = 0 to x = 2 is rotated about the x-axis. Find the volume.
- y² = x².
- V = π∫ x² dx from 0 to 2 = π[x³/3] from 0 to 2.
Answer: 8π/3.
Marks: M1 for π∫ x² dx, A1 for 8π/3.
Example 2. The region under y = √(x + 1) from x = 0 to x = 3 is rotated about the x-axis. Find the volume.
- y² = x + 1.
- V = π∫ (x + 1) dx from 0 to 3 = π[x²/2 + x] from 0 to 3.
- π(9/2 + 3).
Answer: 15π/2.
Marks: M1 for y² = x + 1, A1 for 15π/2.
Example 3. The region between y = x² and the y-axis, from y = 0 to y = 4, is rotated about the y-axis. Find the volume.
- x² = y.
- V = π∫ y dy from 0 to 4 = π[y²/2] from 0 to 4.
Answer: 8π.
Marks: M1 for x² = y, A1 for 8π.
Example 4. The region under y = sin x from x = 0 to x = π is rotated about the x-axis. Find the volume.
- y² = sin²x = (1 − cos 2x)/2.
- V = π∫ (1 − cos 2x)/2 dx from 0 to π = π[x/2 − (sin 2x)/4] from 0 to π.
- π × π/2.
Answer: π²/2.
Marks: M1 for the double angle identity, A1 for π²/2.
Example 5. The region between y = 4 and y = x², from x = 0 to x = 2, is rotated about the x-axis. Find the volume.
- Square each curve: 4² = 16 and (x²)² = x⁴.
- V = π∫ (16 − x⁴) dx from 0 to 2 = π[16x − x⁵/5] from 0 to 2.
- π(32 − 32/5).
Answer: 128π/5.
Marks: M1 for 16 − x⁴, A1 for 128π/5.
Common mistakes
- Forgetting to square y. The formula uses y², not y.
- Leaving out π. V = π∫ y² dx; the π is part of the formula.
- Squaring a sum term by term. (x² + 1)² = x⁴ + 2x² + 1.
- Using (y₁ − y₂)² for a region between curves. Square each curve separately and subtract.
- Using x-limits when rotating about the y-axis. Rotation about the y-axis needs y-limits and x² in terms of y.
How volumes of revolution are examined
Volumes of revolution are part of Pure 4 (WMA14) in Edexcel International A-Level. They are not in the single A-Level Maths specification (9MA0); 9MA0 students meet them in Further Maths instead. Questions typically:
- ask you to show that y² simplifies to a given form
- find the exact volume when a region is rotated about the x-axis
- use a double angle identity for a trig curve
- combine with substitution or parts for a harder integral
Find full volume questions with diagrams and mark schemes in the Pure 4 integration exam questions. Extra volumes to practise are on the Pure 4 integration worksheet, and the integration revision engine can generate more.
For the trig identities, see integrating trig functions. Some volumes also need integration by substitution.
Frequently asked questions
What is the formula for a volume of revolution?
About the x-axis, V = π∫ y² dx between the limits. About the y-axis, V = π∫ x² dy.
Why is y squared in the formula?
Each thin slice is a disc of radius y, and the area of a circle is πr², so each slice contributes πy² times its thickness.
Are volumes of revolution in A-Level Maths?
In Edexcel International A-Level they are in Pure 4. In the Edexcel A-Level Maths 9MA0 specification they are not included; they appear in Further Maths.