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Contents · 0% complete
  1. Learning objectives
  2. Core idea
  3. Key formulae
  4. Worked examples
  5. Explore the graph
  6. Common mistakes
  7. Exam tip
  8. Guided practice
  9. Practice questions
  10. Challenge
  11. Summary
  12. Quick quiz
  13. Same topic, different board
  14. Past paper questions
Learn7367Paper 2 · Compulsory contentFurther calculusVolumes, mean values and improper integrals
Further calculus · Subtopic 1 of 3

Volumes, Mean Values and Improper Integrals

Spin a curve about an axis and the solid's volume is π∫y2 dx\pi\int y^2\,dx (or π∫x2 dy\pi\int x^2\,dy). The mean value of ff on [a,b][a,b] is 1b−a∫abf dx\frac1{b-a}\int_a^bf\,dx. An improper integral has an infinite limit or an infinite integrand: replace it by a limit and see whether that limit exists.

Edexcel 9FM0 Paper 1Edexcel 9FM0 Paper 2AQA 7367 Paper 1AQA 7367 Paper 2≈ 45 minRevision 1 · updated 2026-10-27
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Section 1 of 14

Learning objectives

  • Find volumes of revolution about the xx- and yy-axes.
  • Find the mean value of a function over an interval.
  • Evaluate improper integrals as limits and decide whether they converge.
Pause and think

Can an infinitely long region have a finite area?

Section 2 of 14

Core idea

Start herePaint that fills but can't cover

Spin y=1xy=\frac1x (for x≥1x\ge1) around the xx-axis and you get Gabriel's horn: infinitely long, yet its volume is exactly π\pi. Integrals over infinite ranges can still have finite values, if the integrand fades fast enough.

The big idea

Vx=π∫y2 dxV_x=\pi\int y^2\,dx, Vy=π∫x2 dyV_y=\pi\int x^2\,dy. Mean value 1b−a∫abf dx\frac1{b-a}\int_a^bf\,dx. Improper integrals are limits: ∫a∞f=lim⁡b→∞∫abf\int_a^\infty f=\lim_{b\to\infty}\int_a^bf.

1. Volumes of revolution

V=π∫y2 dxV=\pi\int y^2\,dx
  1. δV≈πy2 δx\delta V\approx\pi y^2\,\delta x
    A thin slice is a disc of radius yy, thickness δx\delta x.
  2. V=lim⁡∑πy2 δx=π∫aby2 dxV=\lim\sum\pi y^2\,\delta x=\pi\int_a^by^2\,dx
    Add the discs.
About the yy-axis use discs of radius xx: π∫x2 dy\pi\int x^2\,dy.
Worked exampley=x2y=x^2 about the yy-axis, 0≤y≤20\le y\le2

Volume.

Goal
VV.
Strategy
Express x2x^2 in terms of yy.
  1. x2=yx^2=y
    From the curve.
  2. V=π∫02y dy=π[y22]02V=\pi\int_0^2y\,dy=\pi\left[\tfrac{y^2}2\right]_0^2
    Discs.
  3. =2π=2\pi
    Evaluate.
Answer2π2\pi.
✓ CheckHalf the cylinder of radius 2\sqrt2, height 22 (4π4\pi): a paraboloid fills exactly half ✓.

2. Mean value

fˉ=1b−a∫abf(x) dx\bar f=\frac1{b-a}\int_a^bf(x)\,dx
where
fˉ\bar f
the height of the rectangle with the same area and width.

3. Improper integrals

∫1∞x−p dx\int_1^\infty x^{-p}\,dx
  1. ∫1bx−pdx=b1−p−11−p\int_1^bx^{-p}dx=\frac{b^{1-p}-1}{1-p}
    For p≠1p\neq1.
  2. p>1: b1−p→0⇒1p−1p>1:\ b^{1-p}\to0\Rightarrow\frac1{p-1}
    Converges.
  3. p<1: b1−p→∞p<1:\ b^{1-p}\to\infty
    Diverges.
  4. p=1: ln⁡b→∞p=1:\ \ln b\to\infty
    Diverges.
Worked example∫01dxx\int_0^1\frac{dx}{\sqrt x}

Evaluate.

Goal
Value.
Strategy
Integrand infinite at 00: limit from the right.
  1. lim⁡a→0+∫a1x−1/2dx\lim_{a\to0^+}\int_a^1x^{-1/2}dx
    Write as a limit.
  2. =lim⁡a→0+(2−2a)=2=\lim_{a\to0^+}\left(2-2\sqrt a\right)=2
    a→0\sqrt a\to0.
Answer22.
✓ CheckHere p=12<1p=\frac12<1, so the integral near 00 converges ✓.
Common misconception"Plugging in ∞\infty" Write the limit explicitly and say what happens to the term in bb.

Summary

  • Disc method about either axis.
  • Mean value: area ÷ width.
  • Improper integrals as limits.
Key words
Converges
The limit exists and is finite.
Diverges
No finite limit.
HistoryGabriel's horn. Evangelista Torricelli (1643) showed that rotating y=1xy=\frac1x for x≥1x\ge1 gives a solid of finite volume π\pi but infinite surface area: you could fill it with paint but not paint it.
Fun factAverage temperature. Weather services report the mean value of the temperature function over a day: literally 124∫024T(t) dt\frac1{24}\int_0^{24}T(t)\,dt.
Section 3 of 14

Key formulae

VolumesVx=π∫aby2 dx,Vy=π∫cdx2 dyV_x=\pi\int_a^by^2\,dx,\qquad V_y=\pi\int_c^dx^2\,dy
Mean valuefˉ=1b−a∫abf(x) dx\bar f=\frac1{b-a}\int_a^bf(x)\,dx
Improper integral∫a∞f(x) dx=lim⁡b→∞∫abf(x) dx\int_a^\infty f(x)\,dx=\lim_{b\to\infty}\int_a^bf(x)\,dx
TrickPowers of 1/x. ∫1∞dxxp\int_1^\infty\frac{dx}{x^p} converges exactly when p>1p>1; ∫01dxxp\int_0^1\frac{dx}{x^p} converges exactly when p<1p<1.
Section 4 of 14

Worked examples

Example 1 · Improper integral0 / 2 steps
Evaluate ∫1∞dxx2\int_1^\infty\frac{dx}{x^2}.
    Example 2 · About the y-axis0 / 2 steps
    y=x2y=x^2, 0≤y≤20\le y\le2, rotated about the yy-axis. Find the volume.
      Section 5 of 14

      Explore the graph

      Area under 1/x² out to b
      Loading the interactive graph…
      Try this
      1. The horizontal line shows the area 1−1b1-\frac1b from 11 to bb.
      2. As bb grows, it creeps up to 11 but never passes it.

      ∫1bdxx2=1−1b→1\int_1^b\frac{dx}{x^2}=1-\frac1b\to1.

      In the real worldEscape velocity. The work to escape a planet is ∫R∞GMmr2 dr=GMmR\int_R^\infty\frac{GMm}{r^2}\,dr=\frac{GMm}R: an improper integral with a finite answer.
      Section 6 of 14

      Common mistakes

      IncorrectV=π∫y dxV=\pi\int y\,dx
      Correctπ∫y2 dx\pi\int y^2\,dx
      Each slice is a disc of area πy2\pi y^2.
      IncorrectSubstituting ∞\infty directly
      CorrectUse a limit
      ∞\infty is not a number; show the limit.
      Incorrect∫1∞dxx\int_1^\infty\frac{dx}x converges
      CorrectIt diverges
      ln⁡b→∞\ln b\to\infty.

      Infinitely long region

      Area: 1. Exactly 1.

      Section 7 of 14

      Exam tip

      Exam tipWrite the limit. Mark schemes give a mark for writing lim⁡b→∞\lim_{b\to\infty} and showing what happens to the term in bb.
      Section 8 of 14

      Guided practice

      SA-FP-VI-G012 marksL1 FoundationCalculatorVolumes
      The line y=xy=x for 0≤x≤30\le x\le3 is rotated 2π2\pi about the xx-axis. Find the volume (to 3 s.f.).

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      Salam Academy Original Practice · Proposed mark scheme
      Section 9 of 14

      Practice questions

      SA-FP-VI-0012 marksL2 StandardNon-calculatorImproper integrals
      Evaluate ∫1∞dxx2\displaystyle\int_1^\infty\frac{dx}{x^2}.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-FP-VI-0022 marksL2 StandardNon-calculatorImproper integrals
      Evaluate ∫01dxx\displaystyle\int_0^1\frac{dx}{\sqrt x}.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-FP-VI-0032 marksL2 StandardNon-calculatorMean value
      Find the mean value of x2x^2 over 0≤x≤30\le x\le3.

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      Salam Academy Original Practice · Proposed mark scheme
      SA-FP-VI-0042 marksL2 StandardCalculatorVolumes
      y=xy=\sqrt x, 0≤x≤40\le x\le4, is rotated about the xx-axis. Find the volume (to 3 s.f.).

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      Salam Academy Original Practice · Proposed mark scheme

      More questions on every skill in this subtopic are in the question bank under Practise.

      Section 10 of 14

      Challenge

      SA-FP-VI-0053 marksL3 ExamCalculatorVolumes
      The curve y=x2y=x^2, 0≤y≤20\le y\le2, is rotated about the yy-axis. Find the volume (to 3 s.f.).

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      Salam Academy Original Practice · Proposed mark scheme
      Section 11 of 14

      Summary

      • π∫y2dx\pi\int y^2dx or π∫x2dy\pi\int x^2dy.
      • Mean value 1b−a∫f\frac1{b-a}\int f.
      • Improper: limits; check convergence.
      Section 12 of 14

      Quick quiz

      Q1 · SA-FP-VI-Q011 markL1 FoundationNon-calculatorImproper integrals
      Which integral diverges?
      Choose one option

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      Salam Academy Original Practice · Proposed mark scheme
      Section 13 of 14

      Same topic, different board

      Specification
      7367 A-level Further Mathematics
      Where
      Paper 1 and 2 · Further calculus
      Expected
      Volumes of revolution, mean values and improper integrals.
      Calculator
      Permitted on all papers.
      Typical style
      AQA mixes short multiple-choice questions with longer structured ones.
      Open in
      7367 · Paper 17367 · Paper 2
      Section 14 of 14

      Past paper questions

      No past paper questions are tagged to this subtopic yet. We add papers every week.

      Browse past papers

      Salam Academy original content · Content status: reviewed and published. Syllabus mapping: AQA 7367, First teaching 2017 (official specification).