Maths › Further calculus › Volumes of revolution
Volumes of revolution
Spin a curve round an axis and it sweeps out a solid. Slice the solid into discs and integration adds them up.
Builds on Definite integrals and areas and Areas and the limit of a sum.
IN THIS TOPIC
- Use V = π∫y² dx for rotation about the x-axis, with the right limits.
- Swap to V = π∫x² dy for rotation about the y-axis.
- Subtract solids for a region between two curves, and handle parametric curves.
- Check answers against known solids such as cones.
COMMON MISCONCEPTION
For a region between two curves, subtract the curves first and square after: V = π∫(youter − yinner)² dx.
Slicing into discs
Rotate the region under y = f(x) about the x-axis and every vertical strip becomes a thin disc of radius y and thickness dx. A disc has volume πy² dx, and integration stacks the discs:
Neither volume of revolution is printed in the booklet, so both belong in your memory. The same slicing sideways handles rotation about the y-axis, giving V = π∫x² dy with limits read off the y-axis. The squared radius is what carries the third dimension, and it is the radius of each disc that gets squared, a point that matters as soon as two curves are involved.
WORKED EXAMPLE
A curve made solid
The region under y = √x from x = 0 to 4 is rotated fully about the x-axis. Find the volume.
V = π∫y² dx = π∫x dx from 0 to 4.
= π[x²/2] = π × 8 = 8π.
Notice how squaring tamed the surd before any integration happened. Rotation about the x-axis often simplifies exactly that way.
Sanity checks and the y-axis
Rotating y = 2x from x = 0 to 3 should make a cone, and it does. V = π∫4x² dx = 36π, matching πr²h/3 with r = 6 and h = 3. Known solids are free marking checks, so use them whenever the curve is a line.
WORKED EXAMPLE
About the y-axis instead
The region between y = x², the y-axis and y = 4 is rotated about the y-axis. Find the volume.
The radius is now x, and x² = y on the curve.
V = π∫x² dy = π∫y dy from 0 to 4 = π × 8 = 8π.
Same number as the first example by coincidence. The setup is the mirror image, with limits on y and radius x.
GUIDED PRACTICE
A cone by integration
By rotating y = 3x between x = 0 and x = 2 about the x-axis, verify the cone volume formula for r = 6, h = 2.
Show the working
V = π∫9x² dx from 0 to 2 = π[3x³] = 24π.
The formula gives πr²h/3 = π × 36 × 2/3 = 24π.
The two answers agree, which is exactly what the disc method promises.
Hollow solids, and parametric curves
A region trapped between two curves spins into a solid with a hole. Compute the outer solid and subtract the inner one, which amounts to π∫(youter² − yinner²) dx over the same limits. Rotating the region between y = x and y = x² from 0 to 1 about the x-axis gives π∫(x² − x⁴) dx = π(1/3 − 1/5) = 2π/15. Squaring each curve separately matters; the square of a difference is not the difference of the squares.
WORKED EXAMPLE
A parametric volume
The curve x = t², y = t³ for 0 ≤ t ≤ 1 is rotated about the x-axis. Find the volume generated.
Change the variable in π∫y² dx. Here dx = 2t dt.
V = π∫ t⁶ × 2t dt from t = 0 to 1 = 2π∫ t⁷ dt.
= 2π[t⁸/8] = π/4.
Convert the limits to t values at the same moment you convert dx. Leaving x-limits on a t-integral is the standard slip.
ASSESSMENT FOCUS
- Square before integrating. The classic error is integrating y and squaring afterwards.
- About the y-axis everything flips. Express x² in terms of y and use y-limits.
- Leave answers as exact multiples of π unless the question asks for decimals.
- For a region between two curves, subtract the two solids. Squaring the gap between them is wrong.
CHECK YOURSELF
The region under y = 2x from x = 0 to 3 is rotated about the x-axis. Find the volume, and name the solid.
Show a hint
π∫4x² dx, then compare with πr²h/3.
Show the answer
V = π∫4x² dx from 0 to 3 = π[4x³/3] = 36π. The solid is a cone with base radius 6 and height 3, and πr²h/3 gives the same 36π.
About the x-axis, V = π∫y² dx between x-limits. Square first, integrate second.
About the y-axis, V = π∫x² dy between y-limits, with x² rewritten in terms of y.
Parametric curves: substitute for y² and dx, and change the limits to t.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
Or read them with their worked answers on the volumes of revolution questions page.
CHECK YOUR PROGRESS
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- Use V = π∫y² dx for rotation about the x-axis, with the right limits.
- Swap to V = π∫x² dy for rotation about the y-axis.
- Subtract solids for a region between two curves, and handle parametric curves.
- Check answers against known solids such as cones.
Open the full revision checklist to see every objective in the course in one place.