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Volumes of revolution and improper integrals (IB Math AA HL)
Updated October 2026
Quick answer: Rotating about the x-axis: V = π∫ y² dx between x-limits. Rotating about the y-axis: V = π∫ x² dy, so write x in terms of y and use y-limits.
V = π ∫ y² dx (about the x-axis) or V = π ∫ x² dy (about the y-axis). HL AA extends to improper integrals with limits at infinity.
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About the x-axis: V = π∫ₐᵇ y² dx. About the y-axis: V = π∫ x² dy. Both are in the formula booklet.
For the y-axis, rearrange to make x the subject, square it, and change the limits to y-values.
For the region between two curves, V = π∫(y₁² − y₂²) dx: square each curve first, then subtract. π∫(y₁ − y₂)² dx is wrong.
Improper integrals: replace ∞ by b, integrate, then find the limit as b → ∞. The volume is finite only if that limit exists.
On Paper 2 the GDC can evaluate the integral; on Paper 1 give the exact answer in terms of π.
Worked example
The region R is bounded by y = √x, the y-axis and the line y = 2. Find the volume when R is rotated through 2π about the y-axis, and the volume when the region under y = √x from x = 0 to x = 4 is rotated through 2π about the x-axis.
About the y-axis: x = y², so x² = y⁴, with 0 ≤ y ≤ 2.
A calculator is allowed on AA Paper 2 and Paper 3, not on Paper 1. Keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50, with a checked worked example:
How do you find a volume of revolution about the y-axis?
Rearrange the curve to x = g(y), then V = π∫ x² dy with the limits as y-values.
Where can I practise volumes of revolution and improper integrals questions?
On the Advanced Integration questions page: 52 exam-style HL AA questions with full mark schemes, plus free step-by-step worked examples.
Can I use my calculator for volumes of revolution and improper integrals?
In HL AA, Paper 1 is non-calculator and the other papers allow a GDC, so learn the exact method as well as the calculator one. The Numerical derivative and integral guide gives the keystrokes for the TI-84 Plus CE, TI-Nspire CX II and Casio fx-CG50.