Volumes of Revolution (AI HL)
Spin a 2D region a full turn about an axis and it sweeps out a solid - this page is about finding that solid's volume with a single integral. The idea is a small step up from area: square the boundary function first, then integrate. It's part of the broader Integration topic.
11 questions on this sub-topic.
The formula (and which way round to use it)
Covered under IB syllabus reference AHL5.12. The formula itself is in the booklet - the skill being tested is choosing the correct axis version and setting up \(y^2\) (or \(x^2\)) correctly before you integrate.
Volume of revolution
\[V=\pi\int_a^b y^2\,dx\]
Rotating the region under \(y=f(x)\) a full turn about the \(x\)-axis sweeps out this volume; swap \(x\) and \(y\) to rotate about the \(y\)-axis instead.
✓ In the formula bookletChoosing \(dx\) or \(dy\)
\[x\text{-axis} \Rightarrow \pi\!\int y^2\,dx \qquad y\text{-axis} \Rightarrow \pi\!\int x^2\,dy\]
The variable you integrate with respect to always matches the axis being rotated about - mixing this up is the single most common setup error.
Need help evaluating \(\int y^2\,dx\) on your calculator? See the parent Integration page.
Worked examples
The region bounded by \(y = x^2\), the \(x\)-axis and the lines \(x = 0\) and \(x = 2\) is rotated \(360^\circ\) about the \(x\)-axis.
(a) Write down an integral for the volume.
(b) Calculate the exact volume.
Worked solution
(a) \(V = \pi \displaystyle\int_0^2 x^4\, dx\) A1
(b) \(V = \pi \left[\dfrac{x^5}{5}\right]_0^2\) M1
\(= \pi \cdot \dfrac{32}{5} = \dfrac{32\pi}{5}\) A1
The region between \(y = \sin x\) and the \(x\)-axis from \(x=0\) to \(x=\pi\) is rotated \(2\pi\) about the \(x\)-axis.
(a) Find the volume, giving your answer to 3 significant figures.
(b) Give the exact value of \(V\) in terms of \(\pi.\)
Worked solution
(a) \(V = \pi\int_0^{\pi} \sin^2 x\,dx.\) M1 A1
Evaluating, \(\int_0^{\pi}\sin^2 x\,dx = \tfrac{\pi}{2},\) so \(V = \tfrac{\pi^2}{2}\) M1
\(\approx 4.93.\) A1
(b) \(V=\dfrac{\pi^2}{2}\) exactly (from the unrounded integral). A1
Common mistakes
- Forgetting to square the function. The formula needs \(y^2\), not \(y\) - skipping the squaring step is the single most common way marks are lost here.
- Integrating with respect to the wrong variable. Rotating about the \(x\)-axis needs \(dx\); rotating about the \(y\)-axis needs \(dy\) (and \(x\) rewritten in terms of \(y\) first) - mixing these up gives a volume for the wrong solid entirely.
- Dropping the \(\pi\) until the very end - or forgetting it altogether. Keep \(\pi\) attached to the integral throughout the working rather than multiplying it in only at the final line, where it's easy to forget.
Ready to practise properly?
10 volumes-of-revolution questions, marked instantly like the real exam.
Quick answers
What is the formula for a volume of revolution?
\(V=\pi\int_a^b y^2\,dx\) for a rotation about the \(x\)-axis, or swap \(x\) and \(y\) to rotate about the \(y\)-axis instead.
Do I integrate with respect to x or y?
Match the variable to the axis of rotation: integrate with respect to \(x\) for a rotation about the \(x\)-axis, and with respect to \(y\) for a rotation about the \(y\)-axis.