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IB Mathematics: Applications and Interpretation HL · AHL 3: Further Integration · AHL 3.8

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Volumes of revolution about the x-axis Problem set A

Sketch the region first. Square the whole function, put the x-limits on the integral and use the calculator. Give answers to 3 significant figures unless told otherwise.

Difficulty: first skills applying them stretch

Skills

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

    0.511.522.53−20−101020xy

  2. The region between the curve \(y=0.5x^{3}\), the x-axis and the lines \(x=0\) and \(x=4\) is rotated 360° about the x-axis. Find the volume of the solid, to 3 significant figures.

    1234−40−30−20−1010203040xy

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

    12345−20−101020xy

  4. The region between the curve \(y=e^{1.5x}\), the x-axis and the lines \(x=0\) and \(x=\ln 4\) is rotated 360° about the x-axis. Find the exact volume of the solid, then give it to 3 significant figures.

    0.20.40.60.811.21.4−10−5510xy

  5. The region between the curve \(y=\frac{4}{x+3}\), the x-axis and the lines \(x=0\) and \(x=5\) is rotated 360° about the x-axis. Find the volume of the solid, to 3 significant figures.

    12345−1.5−1−0.50.511.5xy

  6. The diagram shows the curve \(y=\sqrt{x}\sin\frac{x}{2}\) for \(0\le x\le \pi\). The region between the curve and the x-axis is shaded.

    0.511.522.533.5−2−112xy
    1. (a) Find the area of the shaded region.
    2. (b) The region is rotated 360° about the x-axis. Find the volume of the solid formed.

In context

  1. A trumpet bell is a solid of revolution, with lengths in centimetres. The region between the curve \(y=\frac{8}{x+2}\), the x-axis and the lines \(x=0\) and \(x=6\) is rotated 360° about the x-axis. Find the volume of the solid, to 3 significant figures.

    123456−4−224xy

  2. A horn is modelled by rotating a curve about the x-axis, with lengths in metres. The region between the curve \(y=e^{x/4}\), the x-axis and the lines \(x=0\) and \(x=\ln 25\) is rotated 360° about the x-axis. Find the exact volume of the solid, then give it to 3 significant figures.

    0.511.522.533.5−2−112xy

  3. The side view of a toy rocket is modelled with \(x\) and \(y\) in cm. From \(x=0\) to \(x=4\) the edge is the line \(y=2+\frac{3x}{4}\). From \(x=4\) to \(x=8\) the edge is a quadratic with its maximum point at \((4,\,5)\), which passes through \((8,\,0)\). The rocket is the solid made by rotating the region under the edge 360° about the x-axis. Find its volume, to 3 significant figures.

    2468123456x (cm)y (cm)

Stretch

  1. A bowl is made by rotating a curve about the x-axis, with lengths in centimetres. The region between the curve \(y=a x^{2}\), the x-axis and the line \(x=2\), where \(a>0\), is rotated 360° about the x-axis. The volume of the solid is 90 cm³. Find the value of a.

  2. A glass is made by rotating a curve about the x-axis, with lengths in centimetres. The region between the curve \(y=\frac{6}{x+3}\), the x-axis and the lines \(x=0\) and \(x=3\) is rotated 360° about the x-axis. Show that the volume of the solid is exactly \(6\pi\).

    0.511.522.53−2−112xy

jmaths.xyz · AI HL · AHL 3.8 Volumes of revolution about the x-axis