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AI HL · AHL 3

Further Integration

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AHL 3.8

Volumes of revolution about the x-axis

AHL 3.8

Volumes of revolution about the x-axis

Target: I can explain why rotating a region about the x-axis gives a solid made of thin discs of radius y. (+5 more)

By the end of this section

Key points

The volume formula

  • Rotate the region between a curve y = f(x) and the x-axis through 360° about the x-axis. The solid formed is a solid of revolution.
  • The volume is \(V=\int_{a}^{b}\pi y^{2}\,dx\), where a and b are the x-values at the ends of the region.
  • Each thin slice is a disc. Its radius is y and its thickness is a small change in x, so its volume is about πy² × (thickness). The integral adds up the discs.
  • Sketch the region first. Check which x-values you are using as limits.

Squaring the function

  • Square the whole function before you integrate. The radius is y, so the area of the disc is πy².
  • \(\left(x^{2}+5\right)^{2}=x^{4}+10x^{2}+25\). There is a middle term. Never write \(x^{4}+25\).
  • \(\left(3x^{3}\right)^{2}=9x^{6}\): square the number as well. \(\left(e^{kx}\right)^{2}=e^{2kx}\). \(\left(\frac{k}{x+c}\right)^{2}=\frac{k^{2}}{(x+c)^{2}}\).
  • The calculator can do the squaring for you. Keep the whole function in brackets when you type it.

Answers and limits

  • Give 3 significant figures unless the question asks for an exact answer in terms of π.
  • When a limit is \(\ln a\), \(e^{\ln a}=a\) and \(e^{0}=1\). This often gives an exact answer.
  • A volume has cubic units. Check the units in a context question.
  • A volume is not an area: if you find the integral of y instead of y², you have found the area.
  • If the edge of the solid changes formula part way (a line, then a curve), cut at that x-value. \(V=\pi\int_a^{c}\left(f_1\right)^2dx+\pi\int_c^{b}\left(f_2\right)^2dx\).

On the calculator

Volume of revolution about the x-axis

  • Type the integral with the whole function in brackets and squared: π × ∫ (f(x))² dx.
  • Enter the x-values as the limits. For a ln limit, type ln(5) and so on.
  • Read the volume to 3 significant figures. Multiply by π if you integrated without it.

Screenshot and video to come

Area and volume of the same region

  • Check the calculator is in radian mode for a trigonometric function.
  • Area: integrate y. Volume: integrate π y².
  • Store the function so you can reuse it for both integrals.

Screenshot and video to come

Explore: a solid of revolution

Drag the two limits to choose the region. Turn it about the axis and watch each strip sweep out a disc. Then change the number of discs and compare their total with the exact volume.



Limits
Sum of the discs
Exact volume
The sum of the pieces is

The solid is drawn as seen from a little to one side, so each circle looks like an ellipse. Each disc takes its radius from the middle of its strip.

Turn the region to see the discs.

  1. Set θ = 0, then turn slowly to 360°. What does one strip become? Why is its volume π y² δx and not π y δx?
  2. Set n = 1, then 5, then 20. What happens to the sum of the discs? Predict, then show the exact volume.
  3. Choose y = 2/x about the x-axis with a = 1 and show the exact volume. Drag b until it is 2π, about 6.28. Check your b by solving π∫ (2/x)² dx = 2π from 1 to b.
  4. Choose Washers and turn the region 360°. Each strip now leaves a hole, so its volume is π(outer² − inner²) δx. Make the hole bigger. Does the volume go up or down? Predict, then show the exact volume.
  5. Turn y = x² + 1 about the y-axis. Why are the limits now y-values, and why does the working use x² = y − 1?

Worked examples

Problems like the ones in class. Pick one, try it, then open the working.
Volume of a cubic, with a multiple

The region between the curve \(y=3x^{3}\), the x-axis and the lines \(x=1\) and \(x=2\) is rotated 360° about the x-axis. Find the volume of the solid. Give the exact value in terms of π, then to 3 significant figures.

0.511.522.5−20−101020xy
Working
  1. The formula

    \(V=\pi\int_{1}^{2}y^{2}\,dx\)

    Each thin disc has radius y, so area πy². The limits are x-values.

  2. Square the whole function

    \(y^{2}=(3x^{3})^{2}=9x^{6}\)

    Square the 3 and the x³.

  3. Calculator

    From the calculator: \(\pi\int_{1}^{2}9x^{6}\,dx = 513\)

    Exact value: \(\frac{1143\pi}{7}\).

Video walkthrough coming
Volume when y = x² + c

The region between the curve \(y=x^{2}+5\), the x-axis and the lines \(x=0\) and \(x=2\) is rotated 360° about the x-axis. Find the volume of the solid to 3 significant figures.

0.511.52−10−5510xy
Working
  1. The formula

    \(V=\pi\int_{0}^{2}\left(x^{2}+5\right)^{2}dx\)

    Put the whole function in brackets before squaring.

  2. Expand

    \(\left(x^{2}+5\right)^{2}=x^{4}+10x^{2}+25\)

    There is a middle term: 2 × x² × 5. You can also leave the brackets in for the calculator.

  3. Calculator

    From the calculator: \(\pi\int_{0}^{2}\left(x^{2}+5\right)^{2}dx = 261\)

    3 significant figures.

Video walkthrough coming
Volume of an exponential with a ln limit

The region between the curve \(y=e^{x/2}\), the x-axis and the lines \(x=0\) and \(x=\ln 5\) is rotated 360° about the x-axis. Find the exact volume of the solid, then give it to 3 significant figures.

0.511.5−2−112xy
Working
  1. Square the function

    \(y^{2}=\left(e^{x/2}\right)^{2}=e^{x}\)

    Powers multiply: 2 × x/2 = x.

  2. The integral

    \(V=\pi\int_{0}^{\ln 5}e^{x}\,dx=\pi\left[e^{x}\right]_{0}^{\ln 5}\)

    The upper limit is ln 5, so e to the upper limit is 5.

  3. Exact value

    \(V=\pi(5-1)=4\pi\)

    e⁰ is 1.

  4. Calculator check

    From the calculator: \(\pi\int_{0}^{\ln 5}e^{x}\,dx = 12.6\)

    3 significant figures.

Video walkthrough coming
Volume of a reciprocal curve

The region between the curve \(y=\frac{5}{x+2}\), the x-axis and the lines \(x=0\) and \(x=3\) is rotated 360° about the x-axis. Find the volume of the solid, in terms of π and to 3 significant figures.

0.511.522.533.5−3−2−1123xy
Working
  1. Square the function

    \(y^{2}=\frac{25}{(x+2)^{2}}\)

    Square the 5 and the bracket.

  2. The integral

    \(V=\pi\int_{0}^{3}\frac{25}{(x+2)^{2}}\,dx\)

    Enter it as it stands. The calculator handles the bracket.

  3. Calculator

    From the calculator: \(\pi\int_{0}^{3}\frac{25}{(x+2)^{2}}\,dx = 23.6\)

    Exact value: \(\frac{15\pi}{2}\).

Video walkthrough coming
Area and volume with the calculator

The diagram shows the curve \(y=\sqrt{x}\sin 2x\) for \(0\le x\le\frac{\pi}{2}\). (a) Find the area of the shaded region. (b) The region is rotated 360° about the x-axis. Find the volume of the solid. Give both answers to 3 significant figures.

0.511.5−1−0.50.51xy
Working
  1. Radians

    Calculator in radian mode.

    The limits and the sine are in radians.

  2. (a) Area

    From the calculator: \(\int_{0}^{\pi/2}\sqrt{x}\sin 2x\,dx = 0.861\)

    The area is the integral of y.

  3. (b) Volume

    \(V=\pi\int_{0}^{\pi/2}\left(\sqrt{x}\sin 2x\right)^{2}dx=\pi\int_{0}^{\pi/2}x\sin^{2}2x\,dx\)

    Square the whole product: (√x)² = x and sin² 2x.

  4. Calculator

    From the calculator: \(\pi\int_{0}^{\pi/2}x\sin^{2}2x\,dx = 1.94\)

    3 significant figures.

Video walkthrough coming
An edge in two pieces

The side view of a spinning top is modelled with \(x\) and \(y\) in cm. From \(x=0\) to \(x=2\) the edge is the line \(y=3+\frac{x}{2}\). From \(x=2\) to \(x=6\) the edge is a quadratic with its maximum point at \((2,\,4)\), which passes through \((6,\,0)\). The top is the solid made by rotating the region under the edge 360° about the x-axis. Find its volume, to 3 significant figures.

12345612345x (cm)y (cm)
Working
  1. The quadratic

    \(y=4+a(x-2)^2\), and \(y=0\) at \(x=6\): \(4+16a=0\Rightarrow a=-\frac14\)

    A maximum at (2, 4) gives the vertex form. The point (6, 0) fixes a.

  2. Cut at x = 2

    \(V=\pi\int_0^{2}\left(3+\frac x2\right)^2dx+\pi\int_2^{6}\left(4-\frac{(x-2)^2}{4}\right)^2dx\)

    Each piece has its own formula and its own limits.

  3. Calculator

    From the calculator: first piece \(77.49\), second piece \(107.2\)

    Type each integral with the whole expression in brackets before it is squared.

  4. Add

    \(V=77.49+107.2=184.7\approx185\) cm³

    Exact value: \(\frac{294\pi}{5}\).

Video walkthrough coming