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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 B

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.

Each question matches the question with the same number in set A. Do the ones you got wrong there.

Difficulty: first skills applying them stretch

Skills

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

    0.511.52−15−10−551015xy

  2. The region between the curve \(y=0.25x^{3}\), 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.

    123456−60−40−20204060xy

  3. The region between the curve \(y=x^{2}-5\), the x-axis and the lines \(x=3\) and \(x=6\) is rotated 360° about the x-axis. Find the volume of the solid, to 3 significant figures.

    123456−30−20−10102030xy

  4. The region between the curve \(y=e^{2.5x}\), the x-axis and the lines \(x=0\) and \(x=\ln 2\) is rotated 360° about the x-axis. Find the exact volume of the solid, then give it to 3 significant figures.

    0.10.20.30.40.50.60.70.8−6−4−2246xy

  5. The region between the curve \(y=\frac{9}{x+4}\), 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.

    12345−2−112xy

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

    0.511.5−0.50.5xy
    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 wine glass bowl is a solid of revolution, with lengths in centimetres. The region between the curve \(y=\frac{6}{x+1}\), the x-axis and the lines \(x=1\) and \(x=5\) is rotated 360° about the x-axis. Find the volume of the solid, to 3 significant figures.

    12345−3−2−1123xy

  2. A decorative vase is modelled by rotating a curve about the x-axis, with lengths in decimetres. The region between the curve \(y=e^{x/6}\), the x-axis and the lines \(x=0\) and \(x=\ln 64\) is rotated 360° about the x-axis. Find the exact volume of the solid, then give it to 3 significant figures.

    1234−2−112xy

  3. The side view of a wooden finial is modelled with \(x\) and \(y\) in cm. From \(x=0\) to \(x=3\) the edge is the line \(y=2.5+\frac{x}{2}\). From \(x=3\) to \(x=9\) the edge is a quadratic with its maximum point at \((3,\,4)\), which passes through \((9,\,0)\). The finial is the solid made by rotating the region under the edge 360° about the x-axis. Find its volume, to 3 significant figures.

    246812345x (cm)y (cm)

Stretch

  1. A candle mould is made by rotating a curve about the x-axis, with lengths in centimetres. The region between the curve \(y=a x^{3}\), the x-axis and the line \(x=2\), where \(a>0\), is rotated 360° about the x-axis. The volume of the solid is 150 cm³. Find the value of a.

  2. A lamp base is made by rotating a curve about the x-axis, with lengths in centimetres. The region between the curve \(y=\frac{8}{x+1}\), the x-axis and the lines \(x=1\) and \(x=3\) is rotated 360° about the x-axis. Show that the volume of the solid is exactly \(16\pi\).

    0.511.522.53−4−224xy

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