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IB Mathematics: Applications and Interpretation HL · AHL 3: Further Integration · AHL 3.9

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Volumes of revolution about the y-axis Problem set A

Sketch the region first. Write x² in terms of y, convert the limits to y-values, then use the calculator. Give answers to 3 significant figures unless told otherwise.

Difficulty: first skills applying them stretch

Skills

  1. The part of the curve \(y=2x^{2}+3\) for \(0\le x\le 3\) is rotated 360° about the y-axis. Find the volume of the solid enclosed by the curve, the y-axis and the horizontal lines through the two ends of the curve.

    0.511.522.533.55101520xyy = 2x² + 3

  2. The part of the curve \(y=2+\ln x\) for \(1\le x\le 4\) is rotated 360° about the y-axis. Find the volume of the solid enclosed by the curve, the y-axis and the horizontal lines through the two ends of the curve.

    12341234xy

  3. The part of the curve \(y=\tan x\) for \(0\le x\le 1\) is rotated 360° about the y-axis. Find the volume of the solid enclosed by the curve, the y-axis and the horizontal lines through the two ends of the curve. Use radians.

    0.20.40.60.811.20.511.5xyy = tan x

  4. The part of the curve \(y=\frac{2}{x-3}\) for \(4\le x\le 8\) is rotated 360° about the y-axis. Find the volume of the solid enclosed by the curve, the y-axis and the horizontal lines through the two ends of the curve.

    2468100.511.522.53xyy = 2/(x − 3)

  5. The diagram shows the region bounded by the curve \(y=\frac{2}{x}\), the y-axis and the lines \(y=1\) and \(y=4\).

    0.511.522.53123456xyy = 2/x
    1. (a) Find the area of the region.
    2. (b) The region is rotated 360° about the y-axis. Find the exact volume of the solid formed.

  6. The region between the curve \(y=\sqrt{x^{2}+4}\), the y-axis and the line \(y=5\) is shown in the diagram. It is rotated 360° about the y-axis. Find the volume of the solid, as a multiple of π.

    123456123456xyy = √(x² + 4)

In context

  1. A bowl is made by rotating a curve about the y-axis, with lengths in centimetres. The part of the curve \(y=\frac{x^{2}}{2}+2\) for \(0\le x\le 4\) is rotated 360° about the y-axis. Find the volume of the solid enclosed by the curve, the y-axis and the horizontal lines through the two ends of the curve.

    12345246810xyy = x²/2 + 2

  2. A vase is made by rotating a curve about the y-axis, with lengths in centimetres. The part of the curve \(y=3+\ln x\) for \(1\le x\le 5\) is rotated 360° about the y-axis. Find the volume of the solid enclosed by the curve, the y-axis and the horizontal lines through the two ends of the curve.

    12345612345xy

  3. A water jug is made by rotating the curve \(y=9\ln(x+1)\), for \(0\le y\le 18\), 360° about the y-axis. Lengths are in cm.

    12345675101520x (cm)y (cm)
    1. (a) Find the capacity of the jug in litres.
    2. (b) The jug holds 300 cm³ of water. Find the depth of the water.

Stretch

  1. The diagram shows the region bounded by the curve \(x=4-(y-2)^{2}\) and the line \(x=3\). The region is rotated 360° about the y-axis. Find the volume of the solid, as a multiple of π.

    12345−112345xy

  2. A paperweight is made by rotating a curve about the y-axis. The part of the curve \(y=x^{2}+1\) for \(0\le x\le k\) is rotated 360° about the y-axis. The solid enclosed by the curve, the y-axis and the horizontal lines through the ends of the curve has volume 50 cm³. Find the value of k.

jmaths.xyz · AI HL · AHL 3.9 Volumes of revolution about the y-axis